Thus far, the books and articles I have read dealing with spin groups $\mathbf{Spin}(n)$ and $\mathbf{Spin}(p,q)$ consider them only in terms of either Clifford algebras or topologically as the double covers, respectively, of the special orthogonal groups $\mathbf{SO}(n)$ and $\mathbf{SO}(p,q)$. The spin groups are Lie groups, and I'm aware that there are exceptional isomorphisms between some of them and a few matrix groups. But my question is in general, can $\mathbf{Spin}(n)$ and $\mathbf{Spin}(p,q)$ be viewed just in terms of matrices and/or linear operators? If so, what are some good references that might be able to help in answering this question?



Spin groups in terms of matrices and/or linear operators
Spin groups are algebraic and so they can be presented as subgroups of general linear group that is defined by a system of algebraic equations. You can find one possible presentation in this [answer](https://mathoverflow.net/a/138179/6818) on MO.


