 Almost complex structure from a decomposable volume form

A locally decomposable complex $n$-form $\Omega$ on a real $2n$-manifold, with $\Omega \wedge \overline{\Omega }$ nowhere zero, determines an almost complex structure. The intrinsic rank-$n$ subbundle $W=\{\alpha \in T^{\ast }M\otimes \mathbb{C}:\alpha \wedge \Omega =0\}$ is locally spanned by the factors of $\Omega$. The nonvanishing condition gives $T^{\ast }M\otimes \mathbb{C}=W\oplus \overline{W}$. Set $J^{\ast }=i$ on $W$ and $J^{\ast }=-i$ on $\overline{W}$. This commutes with conjugation, so comes from a unique real bundle endomorphism with square $-I$.