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?
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13$\begingroup$ The right thing to google is "spin representation", which provides a natural way to view the spin groups as matrix groups. There are a number of ways to construct the spin representations, but normally one views $Spin(n)$ as a subgroup of the Clifford algebra on $n$ generators and constructs a representation of the Clifford algebra on an exterior algebra. Studying these representations carefully yields the exceptional isomorphisms that you alluded to. The wikipedia page on the spin representation has a lot of detail; otherwise check out LawsonMichelsohn. $\endgroup$ – Paul Siegel Nov 11 '18 at 18:29
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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 on MO.