Assume that we have a homogeneous space $M\simeq G/H$, where $G$ acts on $M$ transitively, and $H$ corresponds to the stabilizer of a generic point $x_0$, identified with the trivial coset $eH$.
Suppose that we are lucky enough so that the dimension of $M$ is even, and therefore we can endow the tangent space at the marked point $x_0$ with a nondegenerate skew-symmetric bilinear form $\omega_0$.
I am not very familiar with symplectic homogeneous spaces, so I am looking for general arguments or references towards answering the question:
QUESTION: what are the obstructions to left-translate the skew-symmetric bilinear form by transitivity of the action and end up with an honest symplectic structure $\omega$ on $M$?
If $H$ were trivial, and $G$ is compact, this could be done at least in two (a priori) different ways: pick a $G$-invariant metric on $M\simeq G$ by averaging any non-invariant one, then defining $\omega|_g = (g^{-1})^*\omega|_e$. This is a $G$-invariant 2-form on $G$ and it is therefore $\nabla^g$-parallel (Levi-Civita connection). A general argument shows that parallel forms (w.r.t. any torsion-free connection) are closed, so $\omega$ is indeed a symplectic form.
Alternatively, $G$ being a Lie group, it is parellalizable, so we are tasked with endowing the trivial even-rank tangent bundle with a skew-symmetric bilinear form, which is easily done.
Motivation and actual problem I want to solve: I want to be able to see the Kähler structure of the "Siegel Upper Half Space" $\frac{Sp(2n,\mathbb{R})}{U(n)}$ by identifying first its symplectic form and then its complex structure.