Let $V$ be a finite dimensional vector space over a field $F$. Consider the bilinear map $End(V) \times End(V) \rightarrow End(V)$ given by $(u,v) \rightarrow u \circ v$ and the map associated linear map of tensor productsw $m : End(V) \otimes End(V) \rightarrow End(V)$.
I am interested in how we can identify an element of the tensor product $End(V)^* \otimes End(V)^* \otimes End(V)$ with the map $m$ which apperently is just another way to describe standard multiplication of matrices. In any case the question can be formulated as follows:
How do we identify $m$ with an element $u^* \otimes v^* \otimes w \in End(V)^* \otimes End(V)^* \otimes End(V)$?