Consider two smooth manifolds $M_1, M_2$ with symplectic forms $\omega_1, \omega_2$. The following excerpt is from Cannas da Silva, Lectures on Symplectic Geometry:
Concerning the last line. For notational convenience, write $\omega = \theta + \eta$. For $n=1$, we get $$ \omega^2 = (\theta + \eta) \wedge (\theta + \eta) = \theta^2 + 2 (\theta \wedge \eta) + \eta^2, $$ since $\theta \wedge \eta = \eta \wedge \theta$. But what happens with the $\eta^2$ and $\theta^2$, why do they vanish? Similarly, one finds, if I am not mistaken, for $n = 2$, $$ \omega^{4} = \theta^4 + 4 \theta^3 \wedge \eta + 6(\theta^2 \wedge \eta^2) + 4 \eta^3 \wedge \theta + \eta^4. $$ (The resemblance with the binomial formula is obvious. But again, the middle term is what we need to get, but what happens with others?)