Let $V$ and $W$ be two $k$-vector spaces of dimension $n$ and let $\circ :V \times W \to k$ be a $k$-bilinear pairing that is nonsingular.
If $\{v_1,..,v_n \}$ is a basis for $V$, how can I see that I can choose a basis $\{ w_1,..,w_n \}$ for $W$ such that $v_i \circ w_j = \delta_{ij}$?
As the pairing is nonsingular, then by definition there is an isomorphism $\varphi \colon V \simeq W^*$ obtained by sending $v \in V$ to the linear functional $\varphi_v \colon w \mapsto v \circ w$. In particular, $\{ \varphi_{v_1},\ldots,\varphi_{v_n} \}$ is a basis of $W^*$. The isomorphism $W \simeq W^{**}$ says that each $\varphi_{v_i} \in W^*$ corresponds to a unique vector $w_i \in W$, with the property that $\varphi_{v_i}(w_j) = \delta_{ij}$. These $w_i$'s must form a basis, since they are the image of the basis $\{ v_1,\ldots,v_n \}$ under the isomorphism $V \stackrel{\varphi}{\simeq} W^* \simeq W$.