= Solution
Odd-dimensional <spheres> cannot carry a <symplectic form>, by even-dimensionality. For $S^{2m}$ with $m\geq2$, the second <de Rham cohomology> group vanishes. Any hypothetical <symplectic form> would therefore be $\omega=d\alpha$. The <symplectic cohomology obstruction> gives
$$
\int_{S^{2m}}\omega^m
=\int_{S^{2m}}d(\alpha\wedge\omega^{m-1})=0
$$
by the <Generalized Stokes theorem>. But the <symplectic orientation> makes $\omega^m$ a positive <volume form>, whose integral on this nonempty compact manifold is positive. This is a contradiction.
The oriented area form on $S^2$ is nondegenerate and automatically closed, since a two-dimensional manifold has no nonzero three-forms. Thus \b[among positive-dimensional spheres, exactly $S^2$ admits a symplectic structure]:
$$
\boxed{n=2\quad\text{for }n\geq1.}
$$
If zero-dimensional <symplectic manifolds> are admitted, $S^0$ also qualifies: its zero two-form is nondegenerate on the zero <tangent spaces>. This is a convention-dependent additional case, not another positive-dimensional example.
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