Is square matrix multiplication canonic?

By canonicity I mean is there any simple set of algebraic rules, like distributivity, associativity, etc. which uniquely characterize standard matrix multiplication $C_{ij}=\Sigma A_{ik}*B_{kj}$. I'm interested specifically in square matrices as I suspect for uniqueness to hold matrix multiplication should respect the operation of taking determinant: $det(AB)=det(A)*det(B)$ and in non-square case $det$ is not defined.

• Are you aware that the multiplication is just composition of functions? – Tobias Kildetoft Mar 7 '16 at 8:54
• Yes, but I would like to have set of rules which would generalize to case of Pfaffian because in that case composition of transformations associated to skew symmetric matrices is not longer skew symmetric as I was just wondering whether there is some universal way of defining multiplication without any reference to associated linear transformations. – user285001 Mar 7 '16 at 8:58