Hi all. Is there any explicit matrix expression for a general element of the special orthogonal group $SO(3)$? I have been searching texts and net both, but could not find it. Kindly provide any references.

3$\begingroup$ For every $n\in\mathbb N$ and any field $k$, there is a birational map $\mathfrak{so}_n\left(k\right)\to \mathrm{SO}_n\left(k\right)$ given by $A\mapsto \left(I_nA\right)^{1}\left(I_n+A\right)$. It is defined for "almost" all $A\in \mathfrak{so}_n\left(k\right)$ (namely, for those without eigenvalue $1$), is injective and is "almost" surjective (meaning that "almost" all elements of $\mathrm{SO}_n\left(k\right)$ are images under this map). Is this anything like what you are searching for? Because I don't think you can do any better. Even for $n=2$, there is nothing explicit in sin and cos! $\endgroup$ – darij grinberg Jul 12 '11 at 17:43

4$\begingroup$ You can write down the general element in terms of sines and cosines of the Euler angles: en.wikipedia.org/wiki/Charts_on_SO(3). Also:en.wikipedia.org/wiki/Euler_angles $\endgroup$ – Jim Bryan Jul 12 '11 at 18:11

$\begingroup$ @darij: "Even for $n=2$ there is nothing explicit in sin and cos!", I don't understand your comment: what about the usual matrix $(\cos\theta$, $\sin\theta$ ;$\sin\theta$, $\cos\theta )$ ? $\endgroup$ – Qfwfq Jul 12 '11 at 18:15

2$\begingroup$ @darij: I am pretty sure it is standard for the notation $\text{SO}(3)$ to refer to the case $k = \mathbb{R}$ only. $\endgroup$ – Qiaochu Yuan Jul 12 '11 at 19:04

2$\begingroup$ @Qiaochu: Yes, I was generalizing. What I mean by "there is nothing explicit in sin and cos" is that $\sin$ and $\cos$ are two functions whose goal IS more or less to parametrize $\mathrm{SO}_2\left(\mathbb R\right)$, so one can wonder what "explicit matrix expression" means, and if it allows $\sin$ and $\cos$, why it doesn't allow any other special function I might think up. $\endgroup$ – darij grinberg Jul 12 '11 at 20:08
Here is the standard quaternion answer: Given $(a,b,c,d)$ such that $a^2+b^2+c^2+d^2=1$, the matrix $$\begin{pmatrix} a^2+b^2c^2d^2&2bc2ad &2bd+2ac \\ 2bc+2ad &a^2b^2+c^2d^2&2cd2ab \\ 2bd2ac &2cd+2ab &a^2b^2c^2+d^2\\ \end{pmatrix}$$ is a rotation and every rotation matrix is of this form. Note that $(a,b,c,d)$ and $(a, b,c,d)$ give the same rotation.
There is a good way to derive the sort of thing you're looking for: use the double cover $SU(2) \to SO(3)$. $SU(2)$ is diffeomorphic to the 3sphere $S^3 \subseteq \mathbb{C}^2$ $$ SU(2) = \left\{ \begin{pmatrix} a & \overline{b} \\ b & \overline{a} \end{pmatrix} : a^2 + b^2 = 1 \right\} $$ Now $SU(2)$ acts on its Lie algebra $\mathfrak{su}_2$ (which is 3dimensional) by conjugation. This action preserves the inner product $$ \langle X, Y \rangle =  \frac12 \mathrm{tr}(XY) = \frac12 \mathrm{tr}(X^*Y)$$ (which is a scalar multiple of the Killing form of $\mathfrak{su}_2$, FYI) and hence this gives a homomorphism $SU(2) \to SO(\mathfrak{su}_2) \simeq SO(3)$. (A priori this gives a map to $O(3)$, but $SU(2)$ is connected so the image lands in $SO(3)$.
Now consider the orthonormal basis for $\mathfrak{su}_2$ given by $$ e_1 = \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}, e_2 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, e_3= \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}, $$ let $$ x = \begin{pmatrix} a & \overline{b} \\ b & \overline{a}\end{pmatrix},$$ and write down the adjoint action of $x$ on $e_1,e_2,e_3$. For instance you get $$ \begin{align} xe_1x^{1} & = \begin{pmatrix} i(a^2  b^2) & 2ia\overline{b} \\ 2i\overline{a} b & i(a^2  b^2) \end{pmatrix} \\ & = (a^2  b^2)e_1 + i(a\overline{b}  \overline{a}b)e_2 + (a \overline{b} + \overline{a}b)e_3. \end{align}$$ This gives you the first column of the matrix representation conjugation by $x$. I'll leave the others to you. But this way you can see where the formulas come from.
This gives exactly David Speyer's answer (possibly modulo some reordering of the basis). His four real numbers $a,b,c,d$ would correspond to my complex numbers $a,b$ via $$a_{mine} = a + i b, \quad b_{mine} = c + id.$$
To give something explicit in sine and cosine,
$$ \left( \begin{array}{ccc} \cos\theta \cos\psi & \cos\phi \sin\psi + \sin\phi \sin\theta \cos\psi & \sin\phi \sin\psi + \cos\phi \sin\theta \cos\psi \\\ \cos\theta \sin\psi & \cos\phi \cos\psi + \sin\phi \sin\theta \sin\psi & \sin\phi \cos\psi + \cos\phi \sin\theta \sin\psi \\\ \sin\theta & \sin\phi \cos\theta & \cos\phi \cos\theta \end{array} \right) $$
Note that three parameters are required. In odd dimension, there is a real eigenvalue. For $SO_n$ this eigenvalue is $+1.$ So there is a fixed vector in some direction. It takes two parameters to specify this point on the unit sphere. The Lie group element is then a rotation around this point. So it takes a third parameter specifying the amount of rotation about that axis.