Let $V$ be an $n$-dimensional vector space over $\mathbb{F}$ with basis $(v_1,\ldots,v_n)$. Inner products on $V$ are in bijection with positive-definite Hermitian matrices $G\in \mathbb{F}^{n\times n}$ by
\begin{align*}
\left(\sum_{j=1}^n a_jv_j,\sum_{i=1}^n b_iv_i\right)_V=b^*Ga.
\end{align*}