Consider finite-dimensional $\mathbb{F}$-vector spaces $V, W, Q$, where $V$ has basis $\{ e_{1}, \ldots, e_{n} \}$, the space $W$ has basis $\{ f_{1}, \ldots, f_{m} \}$, and $Q$ has basis $\{ h_{1}, \ldots, h_{\ell} \}$. Then we can canonically correspond any linear maps $A: V \to W, B : W \to Q$ to matrices \begin{align*} M(A) & = [\pi_{f_i}(e_{j})]_{i, j} , \\ M(B) & = [ \pi_{h_{i}}(f_{j}) ]_{i, j} \end{align*} in such a way that $M(BA) = M(B) M(A)$, where multiplication of matrices is fairly intuitively defined.
My question is: suppose we have a multilinear map $F : V_{1} \times \cdots \times V_{s} \to W$ and all the vector spaces $V_{i}, W$ have distinguished bases; then in the same way that we can define a matrix corresponding to linear map $V \to W$ as a $\dim(W) \times \dim(V)$-tuple of scalars, can we define a $\dim(W) \times \dim(V_{s}) \times \cdots \times \dim (V_{1})$-tuple and a reasonably intuitive "multiplication" of these "multi-matrices" that corresponds with composition?
