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Can the $(n-1)$ power of the hermitian form on a compact complex $n$-fold be $\partial {\bar{\partial}}$-exact?

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Can the square$(n-1)$ power of the hermitian form on a complex $n$-fold be $\partial {\bar{\partial}}$-exact?

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Can the square of the hermitian form be $\partial {\bar{\partial}}$-exact?

Let $\omega$ be the hermitian form for an hermitian metric on a compact complex manifold. Can $\omega^{n-1}$ be $\partial {\bar{\partial}}$-exact? (We know that our hermitian metric must necessarily be balanced and non-Kahler.)