Let $\pi:E\to M$ be a smooth real vector bundle of rank $2n$ over a smooth manifold $M$. Let $J_0:\mathbb{R}^{2n}\to\mathbb{R}^{2n}$ denote the standard complex structure induced by the identification $\mathbb{R}^{2n}\cong\mathbb{C}^n$, and let

\begin{align*} \rho:GL_n(\mathbb{C})\hookrightarrow GL_{2n}(\mathbb{R}) \end{align*}

be the faithful real representation obtained by forgetting complex scalar multiplication. Then smooth almost complex structures on $E$, meaning smooth bundle endomorphisms $J:E\to E$ over $\operatorname{id}_M$ satisfying $J^2=-\operatorname{id}_E$, are naturally in bijection with $GL_n(\mathbb{C})$-reductions of the real frame bundle $\operatorname{Fr}(E)$.