Let $X$ be a complex manifold of complex dimension $n$, and let $h$ be a Hermitian metric on the holomorphic tangent bundle $T^{1,0}X$. Let $(U,z)$ be a holomorphic coordinate chart with coordinates $z=(z_1,\dots,z_n)$, and write the local frame of $T^{1,0}X|_U$ as $e_i=\partial/\partial z_i$. Define the Hermitian matrix-valued function $H:U\to \operatorname{Herm}_n(\mathbb{C})$ by

\begin{align*} H(x)=(h_{i\bar j}(x))_{i,j=1}^n, \end{align*}

where $h_{i\bar j}(x)=h_x(e_i(x),e_j(x))$. With the convention that the Chern curvature matrix is

\begin{align*} \Theta^C=\bar\partial(H^{-1}\partial H) \end{align*}

and the Chern-Ricci form is

\begin{align*} \operatorname{Ric}^C(h)=i\operatorname{tr}\Theta^C, \end{align*}

one has on $U$

\begin{align*} \operatorname{Ric}^C(h)=-i\partial\bar\partial\log\det H. \end{align*}

Equivalently,

\begin{align*} \operatorname{Ric}^C(h)=-i\partial\bar\partial\log\det(h_{i\bar j}). \end{align*}