Orthonormal basis in a built bilinear form

Question

Let $B = \{v_1, \cdots , v_n\}$ a basis of a vector space $V$ over a field $K=\mathbb R$ or $\mathbb C$. We want to know if there exists a bilinear form $f : V × V \to K$ such that $B$ is an orthonormal basis of $V$ with $f$.

We want to build $f$ such that $f(v_i,v_i)=1$ and $f(v_i,v_j)\neq0$ if $i \neq j$. I feel like the statement is true but I don't have any idea of how to show it.

Answer 1

Are you sure you want $f(v_i,v_j)\neq 0$?

Hint: Consider an isomorphism $V\to \mathbf R^n$ or $\mathbf C^n$ mapping $B$ to the standard basis.