= Solution
Let $P=(\omega^{ab})$ be a smooth <antisymmetric matrix> field. It defines a <Poisson bivector> through
$$
\{f,g\}=\omega^{ab}(x)\,\partial_af\,\partial_bg,\qquad f,g\in C^\infty(U).
$$
This <Poisson bracket> is bilinear, antisymmetric and a <derivation> in each argument. It defines a <Poisson manifold> when it also satisfies the <Jacobi identity>. Apply that identity in the form $\{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0$ to the coordinate functions. Since $\{x^a,x^b\}=\omega^{ab}$, it gives
$$
\boxed{J^{abc}:=\omega^{dc}\partial_d\omega^{ab}
+\omega^{db}\partial_d\omega^{ca}
+\omega^{da}\partial_d\omega^{bc}=0.}
$$
This is the <coordinate Jacobi condition for a Poisson bivector>. It is also sufficient: in the <Jacobi identity> for arbitrary functions the terms containing second derivatives cancel by antisymmetry, leaving $J^{abc}(\partial_af)(\partial_bg)(\partial_ch)$. Thus the coordinate condition captures the whole obstruction.
Now suppose the <Poisson bivector> is nondegenerate. Write $S=P^{-1}$, so $S_{ab}\omega^{bc}=\delta_a^c$, and define the <2-form>
$$
\Omega=\frac12 S_{ab}\,dx^a\wedge dx^b.
$$
An overall minus sign in identifying the <symplectic form> depends on the convention for <Hamiltonian vector fields>; it does not affect the closure argument. Differentiating the inverse matrix gives
$$
\partial_dS_{ij}=-S_{ia}(\partial_d\omega^{ab})S_{bj}
=S_{ia}S_{jb}\partial_d\omega^{ab}.
$$
Contract the coordinate <Jacobi identity> with $S_{ia}S_{jb}S_{kc}$. Antisymmetry gives $\omega^{dc}S_{kc}=-\delta_k^d$, and similarly for the other terms, hence
$$
S_{ia}S_{jb}S_{kc}J^{abc}
=-\{\partial_kS_{ij}+\partial_jS_{ki}+\partial_iS_{jk}\}=0.
$$
The cyclic expression is precisely the coefficient of the <exterior derivative> $d\Omega$. Therefore
$$
\boxed{d\Omega=0.}
$$
Since $S$ is antisymmetric and nondegenerate, $\Omega$ is a <symplectic form>. This proves the <closure of the inverse of a nondegenerate Poisson bivector>. The matrix entries $\omega^{ab}$ are scalar functions; the codomain in the source's matrix description should be read as the space of antisymmetric matrices, rather than a vector-valued individual entry.
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