This question occurred to me while thinking on another one here, Name for an operation on matrices?

Can one define in an invariant way a binary operation on vector spaces - let us denote it somehow suggestively by $(V,W)\mapsto V^{\otimes W}$ - with the property$$\dim(V^{\otimes W})=\dim(V)^{\dim(W)}?$$To avoid some complications, let us restrict to the case when both $V$ and $W$ are finite-dimensional.