Symmetric tensors and duals

Question

Let V be a finite dimensional vector space and consider $(Sym^n V^\vee)^\vee$ where $\vee$ denotes thedual, i.e homogenous polynomials in V of degree n. Consider as well $S_n(V)$, consisting of fixed points of the canonical action of $\Sigma_n$ , the nth symmetric group on $V^{\otimes n}$. I am trying toshow that $S_n(V)$ and $(Sym^n V^\vee)^\vee$are isomorphic but no luck. Could someone help me?