Let $V$ be a finite dimensional vector space over a field $F$ and $k\geq 2$ be a positive integer. Let $E$ be a vector space over $F$ and $\pi:V^k\to E$ be a multilinear map such that the following condition is satisfied:
Whenever there is an alternating multilinear map $f:V^k\to W$ into a vector space $W$, there is a unique linear map $\bar f:E\to W$ such that $\bar f\circ \pi=f$.
Question. With the definition above, does it follow that the map $\pi$ is necessarily alternating?
Note. A multilinear map $f:V^k\to W$ is said to be alternating if it vanishes on all tuples which have two distinct entries equal.
EDIT: Thinking about it a bit more, the answer is obviously no. For $E=\otimes^k V$ also fills the bill. So perhaps what I want to ask here is something else, but I cannot formulate it.