  Let $F$ be a non-discrete non-Archimedean locally compact field such that the
characteristic $\mathrm{ch}(F)\ne 2$ and let $\mathcal{O}_F$ be the ring of
integers in $F$. The main results of this paper are Theorem 1.2 that classifies
ergodic probability measures on the space $\mathrm{Skew}(\mathbb{N}, F)$ of
infinite skew-symmetric matrices with respect to the natural action of the
group $\mathrm{GL}(\infty,\mathcal{O}_F)$ and Theorem 1.4, that gives an
unexpected natural correspondence between the set of
$\mathrm{GL}(\infty,\mathcal{O}_F)$-invariant Borel probability measures on
$\mathrm{Sym}(\mathbb{N}, F)$ with the set of
$\mathrm{GL}(\infty,\mathcal{O}_F) \times
\mathrm{GL}(\infty,\mathcal{O}_F)$-invariant Borel probability measures on the
space $\mathrm{Mat}(\mathbb{N}, F)$ of infinite matrices over $F$.
