Let a finite dimensional complex vector space $V$ be given. Let $T^k(V)$ denote the vector space of multilinear maps $V^k\to\mathbb C$.
My original question was going to be as follows: Does there exist a vector space $W$ for which $T^k(V)$ is isomorphic to the vector space $\mathrm{End}(W)$ of linear operators on $W$?
But it occured to me that since any two finite-dimensional, complex vector spaces of the same dimension are isomorphic, and since \begin{align} \dim T^k(V) = (\dim V)^k, \qquad \dim \mathrm{End}(W) = (\dim W)^2 \end{align} The question boils down to whether there exists a vector space $W$ for which \begin{align} (\dim V)^k = (\dim W)^2 \end{align} If I haven't made an error thus far, then it's immediate that if, for example, $\dim V = 2$ and $k = 1$, then no such $W$ exists. In contrast one is guaranteed that $W$ exists at least whenever $\dim V$ is a perfect square.
Am I missing something?
Moreover, for those $T^k(V)$ for which such a $W$ does exist, is there some natural construction that exhibits the isomorphism in an intuitive way? What exactly would a vector space of linear operators that mimicks a vector space of multilinear operators "look like?" In other words, is there some isomorphism in this context that is natural in some sense?