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 $F$.