Classify matrix representations of group

Question

For a permutation group representation $G \hookrightarrow S_n$ we can classifiy all of them by looking at the translations on some subgroup (which are then the point stabilizers). Is there a similar way to classify all linear representations $G \hookrightarrow GL(V)$ for some vector space $V$ (or similar all matrix representations $G \hookrightarrow M_{n\times n}$)?