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?