There are no symplectic structures on spheres with dimension bigger than 2

For a 2-sphere $S^2$ , there is a cannonical symplectic strucuture induced from $\mathbb{R}^3$ :

$\omega_p(u,v)=\left<p,u\times v\right>, p\in S^2, u,v\in T_p S^2$

That is to regard $p$ as a 3-dimensional vector, doing the mixed product with the two tangent vectors, which expressed in the cylindrical coordinate $(\theta,h)$ is $\omega=d\theta\wedge dh$ , however, this symplectic structure $\omega$ cannot be generalized analogously to the higher dimension.

We first start with a compact manifold $M$ of dimension $2n$ without boundaries, which endowed with a symplectic structure $\omega$ , we first claim that $\omega^n=\omega\wedge\cdots\wedge\omega$ is not an exact form, if not, there will exis a $2n-1$ -form $\alpha\in\Omega^{2n-1}(M)$ such that $d\alpha=\omega^n$ , since $\omega^n/n!$ is a volume form, thus via the Stokes’ formula:

$\int_{M}d\alpha=\int_{\partial M}\alpha=\int_{M}\omega^n=0$

a desired contradiction, besides, $\omega^{n-1}$ is also not exact, if not, there will exist $\beta\in \Omega^{2n-3}(M)$ such that $d\beta=\omega^{n-1}$ , hence we have $\omega\wedge d\beta=\omega^n$ , since the symplectic forms are closed, hence we get $d(\omega\wedge\beta)=\omega^n$ , a desired contradiction, by proceeding this argument inductively, all thos $\omega^k$ are not exact, thus the even-ordered de Rham cohomology groups of a compact symplectic manifold are non trivial, i.e $H^{2k}(M)\neq 0$ for all $k$ .

Now, for the sphere $S^{2k}$ , we have $H^0(S^{2k})=H^{2k}(S^{2k})\cong \mathbb{R}$ , equals to 0 for elsewhere, thus it cannot have any symplectic structure unless $k=1$