I am a physicist working in quantum mechanics, and I am trying to learn geometric algebra in order to get a different perspective on the same thing.
In particular, I am interested in the possibility of generating a geometric algebra from the real vector space of $N \times N$ Hermitian matrices, in such a way that the matrix product plays the role of the geometric product. I'm aware that in the $2\times 2$ case, the anticommutation rules of the Pauli matrices are equivalent to the generating condition of a geometric algebra, so the algebra can be constructed from those three vectors. Is it possible to extend this idea to any arbitrary dimension? Specifically, I want to know if I can consider quantum observables (which can be represented by Hermitian matrices) as vectors in some geometric algebra. Thanks in advance.