If $g$ is a non degenerate symmetric bilinear form over a v.s. $V$ of finite dimension I want to find an isomorphism between $V$ and its dual in the more natural way possible.
I thought about defining $\phi:V\to V^{*}$ by $\phi(x)=g_{x}$ where $g_{x}(y)=g(x,y)$. I proved that $\phi$ is linear transformation and is injective but i don't know how to prove that it is surjective.
Now that I'm thinking, could I use the fact that $\dim(V)=\dim(V^{*})$ so then $\phi$ is isomoprhism? Or maybe I defined my $\phi$ in a wrong way