Symplectic form on $\mathbb{R}^4$

Question

How can I show that the $2$–form $\omega = dx_{1} \wedge dx_{2} + dx_{3} \wedge dx_{4}$ on $\mathbb{R}^4$ is not a multiple of a form obtained by projecting onto a $2$-dimensional plane?

Answer 1

If you had $T\colon \Bbb R^4\to\Bbb R^2$ (for some choice of $\Bbb R^2\subset\Bbb R^4$) and $\omega = T^*\phi$ for some $2$-form $\phi$ on $\Bbb R^2$, you would have $\omega\wedge\omega = T^*\phi\wedge T^*\phi = T^*(\phi\wedge\phi) = 0$, but $\omega\wedge\omega\ne 0$.