TITLE: How do we describe standard matrix multiplication using tensor products?
QUESTION [7 upvotes]: 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 $u^* \otimes v^* \otimes w \in End(V)^* \otimes End(V)^* \otimes End(V)$?

REPLY [3 votes]: Depending on taste, one might also have the "motion" go in the opposite direction: with $v\otimes \lambda$ an endomorphism given by $(v\otimes \lambda)(w)=\lambda(w)\cdot v$, for $v,w\in V$ and $\lambda\in V^*$, the multiplication/composition of endomorphisms is
$$
(v\otimes \lambda)\circ (w\otimes \mu) \;=\; \lambda(w)\cdot v\otimes \mu
$$
for $v,w\in V$ and $\lambda,\mu\in V^*$.