Let $X$ be a hyper-Kähler manifold with complex structure $I$ and a metric $g$. It is known that there are two other complex structures $J,K$ on $X$ that generate $S^2$ of possible complex structures for which $g$ is a Kaehler metric. I don't think the choice of $J,K$ are not unique. But in some literature the holomorphic 2-form is given by $ \Omega(*,**)=g(J*,**)+ig(K*,**) $ if we normalize $\Omega$ by $ (-1^{\frac{n(n-1)}{2}})(\frac{i}{2})^{n}\Omega\wedge \overline{\Omega}=\frac{\omega^{n}}{n!}. $ It seems to me that the first equality determines $J,K$ once we fix $I$.
Phrasing again, it is true that $K$ and $J$ are uniquely determined once we fix $I$?