Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ matrices? (Disclaimer : I know very well that $SO(N)$ has a Lie algebra of dimension $N(N-1)/2$ etc. This absolutely not the point of my question.)
To make my problem more understandable, I start with the example of $SO(2)$. All $SO(2)$ matrices $M$ can be written as ($\theta\in [0,2\pi[$)
$$
M=\begin{pmatrix}\cos\theta & \sin\theta\\ -\sin\theta&\cos\theta\end{pmatrix}.
$$
Using the basis of $2\times2$ real matrices
$\sigma_0=\begin{pmatrix}1 & 0\\0&1\end{pmatrix}$, $\sigma_1=\begin{pmatrix}0 & 1\\1&0\end{pmatrix}$,
$\sigma_2=\begin{pmatrix}0 & 1\\-1&0\end{pmatrix}$,
$\sigma_3=\begin{pmatrix}1 & 0\\0&-1\end{pmatrix}$,
one find that
$$M=\cos\theta\;\sigma_0+\sin\theta\;\sigma_2.$$
Clearly, $M$ does not have components along $\sigma_1$ and $\sigma_3$, so the dimension of the smallest linear subspace of $\mathrm{M}_2(\mathbb{R})$ that contains $SO(2)$ is $2$.
How to articulate the reasoning (for the cases $N>2$ in particular) is not completely clear. I guess that we can say that the component along $\sigma_0$ and $\sigma_2$ are independent because $\cos\theta$ and $\sin\theta$ are independent functions (in a functional analysis sense).
Assuming that made sense, we can try to increase $N$. For example, an $SO(3)$ matrix can be written as
$$
M=\left(\begin{matrix}
\cos\varphi\cos\psi - \cos\theta\sin\varphi\sin\psi & -\cos\varphi\sin\psi - \cos\theta\sin\varphi\cos\psi & \sin\varphi\sin\theta\\
\sin\varphi\cos\psi + \cos\theta\cos\varphi\sin\psi & -\sin\varphi\sin\psi + \cos\theta\cos\varphi\cos\psi & -\cos\varphi\sin\theta\\
\sin\psi\sin\theta                                  & \cos\psi\sin\theta                                   & \cos\theta
\end{matrix}\right)\,
$$
with $(\phi,\psi)\in [0,2\pi[^2$ and $\theta\in [0,\pi[$. Now, if I look at each matrix element one by one, they are all independent in a functional sense.$^*$ Does that mean that the ''dimension of the matrix space'' that $SO(3)$ matrices live in is $9$ ? Is there a way to generalize that to arbitrary $N$ ?
In the end, does any of what I wrote above make any sense ?

$^*$ It is slightly more subtle than that for the $\theta$ dependence, because in the end I am interested in doing integral over the Haar measure, which means that one should look at $x=\cos(\theta)\in[-1,1]$. But $x$ and $\sqrt{1-x^2}$ are orthogonal, so all should be fine.
 A: Here is an elementary proof  but less elegant than the one indicated by @Robert Bryant. $\newcommand{\bR}{\mathbb{R}}$ $\DeclareMathOperator{\Mat}{Mat}$ $\DeclareMathOperator{\SO}{SO}$ 
Denote by $\Mat_n(\bR)$ the vector space of $n\times n$ matrices with real entries. Then $\SO(n)\subset \Mat_n(\bR)$ and I assume that you are asking what is the dimension of the smallest vector subspace of $\Mat_n(\bR)$ that contains $\SO(n)$. Denote by $V_n$ this subspace. 
 We want to show that $V_n=\Mat_n(\bR)$ if $n\geq 3$.
For $i<j<k$ let $\Mat_n^{i,j,k}(\bR)$ be the subspace of $\Mat_n(\bR)$ consisting of matrices $A$ such that
$$a_{pq}=0,\;\;\forall p,q\not\in\{i,j, k\}. $$
Note that the span of the union of the subspaces $\Mat_n^{i,j, k}(\bR)$ is $\Mat_n(\bR)$.
We claim that $\Mat_n^{i,j, k}(\bR)\subset V_n$. For simplicity  we assume $i=1, j=2, k=3$. $\newcommand{\be}{\boldsymbol{e}}$ 
For  $R\in \SO(3)$ let $A_R\in\SO(n)$ be the orthogonal transformation defined by $\newcommand{\bone}{\boldsymbol{1}}$ $A_R=R\oplus\bone$ 
Then $A(R,S):=A_R-A_S\in \Mat^{1,2,3}_n(\bR)$, $\forall R,S\in\SO(3)$.
For any  skew-symmetric  $3\times 3$ matrix $X$ we have
$$
A(e^{tX},1)\in \Mat_n^{1,2,3}. $$  Thus
$$ Xe^{tX}\oplus 0=\frac{d}{dt} A(e^{tx},1)\in\Mat^{1,2,3}_n.$$  We deduce that
$$X\oplus 0=\frac{d}{dt}\Big\vert_{t=0}A(e^{tX},1)\in\Mat_n^{1,2,3}.$$
Thus $V_n$ contain all the skew-symmetric matrices in $\Mat^{1,2,3}_n$. Similarly
$$ X^2\oplus 0=\frac{d^2}{dt^2}\Big\vert_{t=0}A(e^{tX},1)\in\Mat_n^{1,2,3}. $$
Thus $X^2\oplus0\in\Mat_n^{1,2,3}\cap V_n$for any skew-symmetric $3\times 3$ matrix $X$.  $\newcommand{\bu}{\boldsymbol{u}}$ 
For any orthornormal basis $\bu_1,\bu_2,\bu_3$ of $\bR^3$ there exists  a skew-symmetric  $3\times 3$ matrix $X=X_{\bu_1,\bu_2}$ such that
$$X^2\bu_1=-\bu_1,\;\;X^2\bu_2=-\bu_2,\;\;X^2\bu_3=0. $$
Now consider the  matrix 
$$Y_{\bu_1}=-\frac{1}{2}\Big( X_{\bu_1,\bu_2}-X_{\bu_2,\bu_3}+X_{\bu_1,\bu_3}\Big). $$
Note that
$$ Y_{\bu_1}\bu_1=\bu_1,\;\;Y_{\bu_1}\bu_2=Y_{\bu_1}\bu_3=0. $$
Define $Y_{\bu_2}$ and $Y_{\bu_3}$ in a similar fashion.   For any $t_1,t_2,t_3\in\bR$ we have
$$(t_1Y_{\bu_1}+t_2Y_{\bu_2}+t_3Y_{\bu_3})\bu_i=t_i\bu_i. $$
Since any symmetric operator on $\bR^3$ is diagonalizable in some orthonormal basis we deduce that $V_n$ contains all the symmetric matrices in $\Mat_n^{1,2,3}$. Thus $\Mat^{1,2,3}_n\subset V_n$.
A: For $n>2$, the span of the matrices in $\mathrm{SO}(n)$ is the full space $M_n(\mathbb{R})$ of $n$-by-$n$ matrices with real entries.
One proof is using representation theory:  If we let $S\subset M_n(\mathbb{R})$ denote the span of $\mathrm{SO}(n)$, then notice that this span is invariant under the action of $\mathrm{SO}(n)\times\mathrm{SO}(n)$ defined by 
$$
(A,B)\cdot C = ACB^{-1}.
$$
Thus, it suffices to show that this representation of $\mathrm{SO}(n)\times\mathrm{SO}(n)$ on $M_n(\mathbb{R})$ is irreducible, since $S$ is clearly not the zero subspace.
When $n>2$, the center of $\mathrm{SO}(n)$ is discrete (it is either the identity (when $n$ is odd) or $\pm$ the identity (when $n$ is even), and the (irreducible) representation of $\mathrm{SO}(n)$ on $\mathbb{R^n}$ has commuting ring isomorphic to $\mathbb{R}$.  Since the above representation of $\mathrm{SO}(n)\times\mathrm{SO}(n)$ on $M_n(\mathbb{R})$ is clearly the tensor product of the irreducible $\mathbb{R}^n$-representations of the two $\mathrm{SO}(m)$-factors of $\mathrm{SO}(n)\times\mathrm{SO}(n)$, it is irreducible.
A: It  is  sufficient to  prove the  statement in the  question only for matrices in the  form $[1]_{1\times 1}\oplus [0]_{k}$, that is  the  rank-1 projections  or $$\begin{pmatrix} 0&\lambda\\ -\lambda&0\end{pmatrix}\oplus[ 0]_k\qquad (*)$$   Assuming that the proof  for  such  matrices is  straithforward  then the  proof  is  completed.
Because  every  matrix is a  sum of  a  symmetric  and  an anti  symmetric  matrix.  Every  symmetric  matrix  is  diagonalizable  via  an orthonormal  matrix  and  every  anti  symmetric  matrix  is  orthonrmally equivalent  to  a  direct  sum of  matrices  in the  form of  $(*)$.
This  would give  us  an  elementary but  less elegant proof than  exisiting  answers, in particular the answer based  on represention  theory, tangent space of $SO(n)$ and extermum points of the unit ball with operator norm.
A: Let $S$ be the span of $SO(N)$ .
Then it's obvious that if $A \in S$ and $D_1, D_2 \in SO(N)$ then $D_1^{-1} A D_2 \in S$ .
Therefore it's enough to show that $B := diag(1,0,...0) \in S$ .
If $N$ is odd and $> 1$ then we can write $$B = \frac{1}{2} (\left[\begin{matrix}1&0\cr0&I_{N-1}\end{matrix}\right] + \left[\begin{matrix}1&0\cr0&-I_{N-1}\end{matrix}\right])$$ .
If $N$ is even and $> 2$ then we can write $$B = \frac{1}{4} (\left[\begin{matrix}1&0&0\cr0&1&0\cr0&0&I_{N-2}\end{matrix}\right] + \left[\begin{matrix}1&0&0\cr0&1&0\cr0&0&-I_{N-2}\end{matrix}\right] + \left[\begin{matrix}1&0&0\cr0&-1&0\cr0&0&E\end{matrix}\right] + \left[\begin{matrix}1&0&0\cr0&-1&0\cr0&0&-E\end{matrix}\right])$$ ,
where we choose $E \in O(N-2)$ with $det E = -1$ .
A: Elementary proof. The linear space $E$ spanned by $SO_n$ is the orthogonal of those matrices $M$ such that $\langle M,Q\rangle:={\rm Tr}(MQ)=0$ for every $Q\in SO_n$. Let $M=SR$ be a polar decomposition, where $S\in Sym_n^+$ and $R\in O_n$. This decomposition is unique with $S\in SPD_n$ if $M$ is non-singular, but in general it exists and might be non-unique. If $R\in SO_n$, then ${\rm Tr}(SQ)=0$ for every $Q\in SO_n$ ; chosing  $Q=I_n$, we have ${\rm tr}\,S=0$, which implies $S=0_n$. If on the contrary $R\in O_n^-$, we have ${\rm Tr}(SQ)=0$ for every $Q\in O_n^-$.  Diagonalize $S$ in an orthogonal basis, the matrix of $D$ eigenvalues satisfies ${\rm Tr}(DQ)=0$ for every $Q\in O_n^-$. Chosing $Q$ the symmetry with respect to hyperplane $x_j=0$, we obtain ${\rm Tr}\,D=2d_j$ for every $j$. There follows $n\,{\rm Tr}\,D=2\,{\rm Tr}\,D$. Whence ${\rm Tr}\,D=0$, $d_j=0$ if $n\ge3$. This yields $M=0_n$, hence $E$ is the full space $M_n$. 
When $n=2$, this gives only $d_1=d_2$ and we recover $M\in O_2^-$.
A: Clearly the span of $SO(n)$ contains the tangent space of $SO(n)$, the antisymmetric matrices. But it also contains the rotations by $\pi$ in any 2-plane, so contains the diagonal matrices with even numbers of $-1$ entries and all other entries $1$. Linear combinations of such easily contain the diagonal matrices. All symmetric matrices are diagonalizable by rotation matrices. So we get all symmetric matrices as well. Hence all $n \times n$ matrices.
A: This argument doesn't work for real orthogonal matrices as far as I can see, but it works for unitary matrices in the complex case, and is somewhat related to Robert Bryant's answer for the real case. It has appeared from time to time in the finite group representaton theory literature, though I'm not sure who first discovered it. 
   For each integer $n > 1,$ there is a nilpotent group $H$ of order $n^{3}$ consisting of monomial matrices ( ie with one non-zero entry in each row and column) with non-zero entries $n$-th roots of unity, such that every non-scalar matrix in $H$ has trace zero (and the scalars in $H$ form the subgroup $Z(H)$ of order $n$). Furthermore, all non-scalar matrices in $H$ have trace zero.
 Let $S$ be a transversal to $Z(H)$ in $H$. Then ${\rm trace}(A\overline{B}^{T}) = 0$ whenever $A,B$ are distinct elements of $S$ (note that $H$ consists of unitary matrices).
 Hence $ S $ is a set of $n^{2}$ mutually orthogonal non-zero matrices in ${\rm M}_{n}(\mathbb{C}),$ with respect to the inner-product $\langle X,Y \rangle = {\rm trace}(X\overline{Y}^{T}).$ Thus $S$ forms a basis of $M_{n}(\mathbb{C}),$ consisting of unitary matrices (of determinant $\pm 1).$
