Let $V$ be a finite dimensional vector space over $\mathbb{C}$. For any set of nonzero vectors $(v_1, v_2, \ldots, v_N )$, we can construct a symmetric tensor
$$ W = \sum_{\sigma\in S_N }v_{\sigma(1)} \otimes v_{\sigma(2)} \cdots \otimes v_{\sigma(N)} . $$
I want to prove that $W\neq 0 $ if the $v$'s are nonzero. The proof I found is as follows. Suppose we have an inner product on $V$. Then consider the inner product of $W$ and the symmetric tensor
$$ S = u \otimes u \otimes \cdots \otimes u . $$
We have
$$ \langle W | S \rangle = N! \prod_{i=1}^N \langle v_i | u \rangle . $$
We can definitely find $u$ such that $\langle v_i | u \rangle \neq 0 $ for all $i$. This is by the common knowledge that a vector subspace is not the union of a finite number of proper subspaces.
But can one find a proof without using the inner product?