Is the convex cone of sums of self tensor products of vectors in a given vector space full-dimensional in the space of symmetric matrices?

Question

Let $V$ be a finite-dimensional real vector space. Let $S_n(V)$ denote the real vector space of all symmetric $n\times n$ matrices with entries in $V$.

Consider the convex cone $C\subset S_n(V) $ generated by taking sums of $v \otimes w := v w^T$, where $v$ and $w$ are column vectors in $V^n$.

Is this convex cone full-dimensional? That is, is every matrix in $S_n(V) $ a linear combination of matrices in $C$?