In this thread Congruence of invertible skew symmetric matrices it is stated that, for any invertible skew-symmetric matrix $G$, there is an invertible matrix $A$ such that \begin{align*} A^\intercal G A = \Omega := \left( \begin{array}{c|c} 0 & I \\ \hline -I & 0 \end{array} \right). \end{align*}
If we assume that $G$ is skew-symmetric and belongs to $SL_{2n}(\mathbb{Z})$, is it then possible to find such an $A \in GL_{2n}(\mathbb{Z})$? Phrased differently, is the action \begin{align*} GL_{2n}(\mathbb{Z}) \curvearrowright \lbrace X \in SL_{2n}(\mathbb{Z}) : \text{$X$ is skew-symmetric} \rbrace, \quad g.X = g^\intercal X g \end{align*}
transitive?