We have a real vector bundle $\pi : E\rightarrow M$ of rank $2n$ over smooth manifold $M$ and $J\in \Gamma(E^*\otimes E)$(set of smooth sections of vector the bundle) is a complex structure on $E$. And suppose $E$ is trivial over an open set $U\subseteq M$. Show that there is a local frame $(e_1,f_1,\dots, e_n, f_n)$ of $E$ over $U$ where $f_i=J(e_i)$.
I have been trying to show this but I have not gotten any advance after I start to think that $e_i$ takes real part and $f_i$ takes imaginary part. I really need help from here. I thank in advance for any help :)
For reference, for any $p\in M$, $J$ should satisfy $J(p)^2=-Id_{E_p}$ where $E_p=\pi^{-1}(\{p\}).$