= Solution
A <symplectic manifold> is a <smooth manifold> $M$ equipped with a <differential two-form> $\omega$ such that
$$
\boxed{d\omega=0,\qquad\omega_p(v,w)=0\text{ for all }w\in T_pM\Longrightarrow v=0.}
$$
The first condition says it is a <closed differential form>; the second says it is pointwise <nondegenerate>. A nondegenerate skew-symmetric matrix has even size, so the dimension is $2n$. Equivalently, $\omega^n$ is a nowhere-zero top-degree <differential form>. It supplies an <orientation> and the <volume form> $\omega^n/n!$. These are the defining conditions on the <symplectic form>.
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