Spin group as an automorphism group Consider the real algebraic group $SO(p,q)$, this is the automorphism group of the vector space $\mathbb{R}^n$ of dimension $n=p+q$ over  $\mathbb{R}$, endowed with the diagonal quadratic form with $p$ pluses and $q$ minuses on the diagonal and with a nonzero skew symmetric  $n$-form.
Now consider the corresponding spinor group $G=Spin(p,q)$. Can one say that it is the group of automorphisms of a some object over $\mathbb{R}$, which is not very difficult to describe?
 A: To expand on my comment to the question, we have the following algebraic construction (I think originally due to  Robert Brown, 'A characterization of spin representations'): 
Let $V$ be a quadratic space over a field $k$ of characteristic $\neq 2$. Attached to this is the Clifford algebra $C=C(V)$: it is equipped with a $\mathbb{Z}/2\mathbb{Z}$-grading $C=C^+\oplus C^-$ and an embedding $V\hookrightarrow C^-$. The general Spin group $GSpin(V)$ consists of units in $C^+$ that preserve $V$ under conjugation. The group $Spin(V)$ is the sub-group of elements that have trivial spinor norm. So, to describe $Spin(V)$ as an automorphism group, it suffices to do so for $GSpin(V)$.
Let $H$ be the graded vector space $C$ viewed as a representation of $GSpin(V)$ via left multiplication: it is also a right $C$-module via right multiplication. Then $GSpin(V)$ clearly lies within the group $U(H)$ of $C$-equivariant, grading preserving automorphisms of $H$. 
Set $E=End(H)$: this is a representation of $GSpin(V)$ via conjugation. Define a bilinear form $\{,\}:E\times E\to k$ by $$\{f,g\}=\frac{1}{2^{dim(V)}}trace(fg).$$ 
Now choose a basis $\{v_i\}$ for $V$, and let $A=(v_i\cdot v_j)_{i,j}$ be the inner product matrix attached to this basis. Set $(b_{i,j})=B=A^{-1}$. Define an endomorphism $\pi:E\to E$ by the formula:
$$\pi(f)(h)=\sum_{i,j}b_{i,j}\{v_i,f\}v_jh.$$
Clearly, the image of $\pi$ is $V\subset E$, where $V$ acts on $H$ via left multiplication. Let $G'\subset U(H)$ be the stabilizer of the endomorphism $\pi$. Then $G'$ preserves $V$ via conjugation, and is therefore contained in $GSpin(V)$. On the other hand, it is not hard to see that $GSpin(V)$ stabilizes $\pi$. 
So we see that $GSpin(V)$ can be described as the group of $C$-equivariant, grading preserving automorphisms of $H$ that also stabilize $\pi$.
A: It seems that you are asking for descriptions of the groups $\mathrm{Spin}(p,q)$ as algebraic groups.  This can certainly be done explicitly for low values of $p$ and $q$, but I don't know a general process that works well when $p$ or $q$ is large.
For low values, the exceptional isomorphisms turn out to provide good descriptions, such as


*

*$\mathrm{Spin}(3) = \mathrm{SU}(2)$

*$\mathrm{Spin}(2,1) = \mathrm{SL}(2,\mathbb{R})$

*$\mathrm{Spin}(4) = \mathrm{SU}(2)\times\mathrm{SU}(2)$

*$\mathrm{Spin}(3,1) = \mathrm{SL}(2,\mathbb{C})$

*$\mathrm{Spin}(2,2) = \mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$

*$\mathrm{Spin}(5) = \mathrm{Sp}(2)$  (i.e., the matrices in $\mathrm{SO}(8)$ that are $\mathbb{H}$-linear)

*$\mathrm{Spin}(4,1) = \mathrm{Sp}(1,1)$  (i.e., the matrices in $\mathrm{SO}(4,4)$ that are $\mathbb{H}$-linear)

*$\mathrm{Spin}(3,2) = \mathrm{Sp}(2,\mathbb{R})$  (i.e., the matrices in $\mathrm{GL}(4,\mathbb{R})$ that are symplectic)

*$\mathrm{Spin}(6) = \mathrm{SU}(4)$ 
and so on.  The higher you go, though, the more complicated the description.  For example, $\mathrm{Spin}(7)$ is the subgroup of $\mathrm{GL}(8,\mathbb{R})$ that preserves a certain $4$-form on $\mathbb{R}^8$.  
By the time you get to $\mathrm{Spin}(10)$, which shows up as a subgroup of $\mathrm{SU}(16)$, it is not at all obvious what extra algebraic structure on $\mathbb{C}^{16}$ you need to characterize it algebraically.  It turns out there is a homogeneous quartic polynomial $Q$ on $\mathbb{C}^{16}$ such that $\mathrm{Spin}(10)$ is the subgroup of $\mathrm{SU}(16)$ that preserves $Q$.
A good place to see this story is Spinors and calibrations by F. Reese Harvey.
A: See 
Groupes Classiques
Baptiste Calmès, Jean Fasel
http://arxiv.org/abs/1401.1992
Propositions 4.5.1.15 and 4.5.1.16.
