= Solution
Let the two oriented surfaces carry positive area forms $\alpha$ and $\beta$, obtainable from oriented Riemannian metrics. A top-degree form on a surface is automatically closed. On their product set
$$
\omega=\pi_1^*\alpha+\pi_2^*\beta.
$$
It is closed, and
$$
\omega^2=2\pi_1^*\alpha\wedge\pi_2^*\beta
$$
is a nowhere-vanishing four-form; the two squared terms vanish because each comes from a two-dimensional factor. Thus \b[the product is symplectic], with no <compactness> assumption needed.
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