Given a nondegenerate, symmetric bilinear form, and linearly independent vectors, find a set of vectors s.t. $g\left(v_{i},w_{j}\right)=\delta_{ij}$

Question

I'm trying to prove the following theorem:

Let $V$ be a $n$-dimensional vector space over $\mathbb{F}$, $g:V\times V\to \mathbb{F}$ a symmetric and non-degenerate bilinear form. If $v_1,\ldots ,v_k \in V$ linearly independent is a set vectors then $\exists w_1 ,\ldots ,w_k \in V$ s.t. $$g\left(v_{i},w_{j}\right)=\begin{cases} 1 & i=j\\ 0 & i\neq j > \end{cases}=\delta_{ij}$$

I tried using induction but it doesn't seem to lead anywhere.