I am studying the Lagrangian Grassmannian to eventually understand the Maslov class and the closely related Maslov index for Heegaard Floer Homology. In the computation of the Lagrangian Grassmannian $L(\mathbb{R}^4,\Omega_0)$, where $\Omega_0$ is the standard symplectic structure, the following fact is used without explanation and I couldn't prove it myself:
Any 2-dimensional subspace $W$ of a 4-dimensional vector space $V$ can be presented as the kernel of a $2$-form $\varphi_W$.
Here $u\in \ker\varphi_W $ if $\varphi_W(u,v) = 0$ for all $v\in V$. How would one construct such a $2$-form? If I just choose a basis of $W$, extend it to $V$, define a $2$-form on the basis elements (so that $W$ is in the kernel) and extend linearly, how can I make sure the kernel won't be strictly bigger than $W$? I would appreciate any hints!