Let $(V, \omega)$ be a finite-dimensional real symplectic vector space, i.e. $\omega : V \times V \to \mathbb{R}$ is a non-degenerate skew-symmetric bilinear map.
A linear subspace $L \subset V$ is called *Lagrangian* if $L = L^\perp$, where $L^\perp = \{v \in V : \omega(v, L) = \{0\}\}$.

Let $U \subset V$ be a linear subspace such that
$$\dim U = \frac{1}{2} \dim V.$$
**Question.** *Why is there always a Lagrangian subspace $L \subset V$ such that $V = U \oplus L$?*

This is easy if $U$ is also Lagrangian, and there are plenty of sources explaining this (we can set $L = I(U)$ where $I$ is a compatible complex structure). But I didn't find any reference explaining this more general fact, although I have seen it used in some research papers. It is used, for instance, for the existence of local Lagrangian bisections in symplectic groupoids. Any hints on the proof would be appreciated.