If $V$ is a finite-dimensional vector space, then $V\otimes V\cong\mathbf{S}^2(V)\oplus\bigwedge^2(V)$.
The first summand on the right is the symmetric part of $V\otimes V$ and the second summand is the anti-symmetric part. This identification of vector spaces is just the decomposition of order-$2$ tensors into their symmetric and anti-symmetric parts. (When $V$ is $1$-dimensional, we simply have $V\otimes V\cong\mathbf{S}^2(V)$.)
Without imposing assumptions on the dimension of $V$, what is the corresponding decomposition for $V\otimes V\otimes V$? There should be an $\mathbf{S}^3(V)$ and a $\bigwedge^3(V)$, but what are the "mixed" terms?
Basically, I'd like a decomposition for $\bigotimes^nV$.
I'm not so interested in the (anti-)symmetrization formulas, i.e. I'm not worried about the specific isomorphism.