I am currently studying normal coordinates on a Kaehler manifolds: Let $h$ be a Kaehler metric on a complex manifold $M$ and let $p \in M$. Let $(z_1,..,z_n)$ be a coordinate chart such that $h$ is a metric in these coordinates. I know that due to the Kaehler condition, we have that $h_{i\bar j}(p)=\delta_{ij}$ after a constant linear change of coordinates $(z'_1, ...,z'_n)$. (In addition, I know that $dh_{i \bar j}=0$). My question is: How does the matrix of $h$ look in the new coordinates $(z'_1, ...,z'_n)$ before I substitute $p$ in to get $h_{i\bar j}(p)=\delta_{ij}$? Is there any kind of change-of-coordinates formula to obtain the matrix $h$ in these new coordinates?
The reason I am asking is because I need to compute the derivatives of the $h_{i\bar j}$ without plugging in $p$.