Given a finite dimensional real vector space $V$ of dimension $2n$, then the space of non-degenerate skew-symmetric forms on it is an open subset of skew-symmetric forms, which is diffeomorphic to the homogeneous space $GL_{2n}(\mathbb{R})/Sp_n(\mathbb{R})$. It is hence made up by two connected components, since $GL_{2n}(\mathbb{R})$ is. Two forms from different components induce opposite orientations on $V$.
Similarly, the space of complex structures is diffeomorphic to $GL_{2n}(\mathbb{R})/GL_{n}(\mathbb{C})$ is made of two connected components.
Given a non-degenerate skew-symmetric form $\omega$ and a complex structure $J$ such that $\omega(J \cdot, J \cdot)= \omega(\cdot, \cdot)$, then what are the possible signatures of the inner product $\omega(\cdot, J \cdot)$?
A first attempt to classify compatible complex structure is to note that any two complex structure are similar, and try to express compatible complex structures as a homogeneous space. For sure if $D$ is symplectic, $DJD^{-1}$ is again compatible, hence a first guess could be that compatible complex structure are the homogeneous space $Sp_{2n}(\mathbb{R})/U(n)$. However, if $J$ is compatible $-J$ is too and it is obtained by conjugating $J$ with an element that reverses the orientation of $V$ and hence that cannot be in $Sp_{2n}(\mathbb{R})/U(n)$. Any help would be appreciated.