I'm trying to make sense of a fundamental definition of Kahler manifolds. In almost any introduction, it is say that given a hermitian metric $g(X,Y)$ and an almost complex structure $J$ such that $J^2=-I$, one can define a complex (1,1)-form $\omega(X,Y)$ = $g(JX,Y)$ = $g(X,JY)$. I understand the effect of $J$ is to rotate a vector i.e. to multiply the components by $i$. It is then stated that the Kahler metric is closed: $d\omega = 0$. What I'm trying to understand is how this statement translates to vectors that lie uniquely in along the real or the imaginary components.
Let's say that the real components have a metric tensor $h$, and the imaginary components have a metric tensor $k$. Is the equation $d\omega = 0$ the same as saying that $dh = -dk$? How would $h$ and $k$ relate to $g$?