I'm starting to study hyperKähler manifolds and I have a question for you. I hope it's not too stupid...

I know that given a Kähler metric $g$ on an hyperKähler manifold $X=(M,I)$ ($M$ is the differential structure and $I$ the complex structure) there are $J$, $K$ complex structures on $M$ with quaternionic relations such that for every $(a,b,c)\in S^2$ the linear combination $aI+bJ+cK$ is a complex structure which makes $g$ Kähler.

The form $\sigma_I := g(J-,-)+ig(K-,-)$ should be a $(2,0)$-form for $X$ which gives the isomorphism $K_X\simeq \mathcal{O}_X$.

But I have trouble proving it: let's take $u$ and $v$ eigenvectors for $I$ (seen as an almost complex structure) with eigenvalue $i$. Then $$\sigma_I(u,v)=g(Ju,v)+ig(Ku,v)=2g(Ju,v)$$ But $\sigma_I$ should be a nowhere zero $(2,0)$-form, so it should be $2g(Ju,v)\neq 0$. Why is that? I just know that $g(J-,-)$ is a $(1,1)$-form for $(M,J)$.