In the case of orthonormal Clifford Algebra, matrix representations of the generators are easy to find. For example, in 3d-Euclidean space,
$$ \frac{1}{2} \left( \sigma_i \sigma_j +\sigma_j \sigma_i \right) = \delta_{i j} $$
The generators are the Pauli matrices.
However, in the case of non-orthonormal algebras, what can and cannot serve as a representation is unclear to me. Suppose the following Clifford Algebra
$$ \frac{1}{2} \left( e_\mu e_\nu+e_\nu e_\mu \right) = g_{\mu\nu} $$
Can we find matrix representations for $e_\nu, e_{\mu}$? If not, what can we use?
Edit: example
For example, one can think of the interval (metric tensor) of general relativity
$$ e_0e_0=g_{00}\\ e_1e_1=g_{11}\\ e_2e_2=g_{22}\\ e_3e_3=g_{33}\\ e_0e_1+e_1e_0=2g_{01}\\ e_0e_2+e_2e_0=2g_{02}\\ e_0e_3+e_3e_0=2g_{03}\\ e_1e_2+e_2e_1=2g_{12}\\ e_1e_3+e_3e_1=2g_{13}\\ e_2e_3+e_3e_2=2g_{23} $$
Is there a valid matrix representation of the generators $e_0,e_1,e_2,e_3$?
Every finite dimensional algebra can be represented as an algebra of matrices.
If you let $V$ be the underlying vector space of a $k$ algebra, then each element $a\in V$ acts by left multiplication on $V$, creating a linear transformation $\lambda_a$ of $V$.
The mapping $a\mapsto \lambda_a\in End_k(V)\cong M_{\dim(V)}(k)$ turns out to be a homomorphism of $k$-algebras, and its image is an algebra of matrices isomorphic to the original algebra.