= Solution
An <almost complex structure> is a smooth bundle endomorphism $J:TM\to TM$ with $J^2=-I$. It is an $\omega$-<compatible almost complex structure> when
$$
\boxed{\omega(Jv,Jw)=\omega(v,w),\qquad\omega(v,Jv)>0\quad(v\ne0).}
$$
These conditions make $g_J(v,w)=\omega(v,Jw)$ a <Riemannian metric>. In particular it is symmetric: invariance and $J^2=-I$ give $\omega(Jv,w)=-\omega(v,Jw)$, and skew-symmetry then gives $g_J(w,v)=g_J(v,w)$. Positivity is the second condition. Moreover $J$ is an isometry for $g_J$. Compatibility does not require that the <almost complex structure> be integrable.
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