Show that after a suitable change of basis, a skew-symmetric nonsingular bilinear form over \(\mathbb{Z}\) can be represented by a matrix consisting of \(2 \times 2\) blocks \(\bigl(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\bigr)\) along the diagonal and zeros elsewhere.
Solution
Let \(L\) be a free abelian group of finite rank with a skew-symmetric nonsingular bilinear form \[ B:L\times L\to\mathbb Z. \] Nonsingularity over \(\mathbb Z\) means the adjoint map \(L\to L^*\) is an isomorphism.
Choose a primitive element \(e\in L\). Since the functional \(B(e,-)\) is primitive in \(L^*\), there exists \(f\in L\) with \[ B(e,f)=1. \] Then the restriction of \(B\) to \(P=\mathbb Ze\oplus\mathbb Zf\) has matrix \[ \begin{pmatrix}0&1\\-1&0\end{pmatrix}. \] For any \(x\in L\), set \[ x'=x-B(x,f)e+B(x,e)f. \] A direct calculation gives \[ B(x',e)=B(x',f)=0. \] Thus \[ L=P\oplus P^\perp. \] The restriction of \(B\) to \(P^\perp\) is again integral, skew-symmetric, and nonsingular. Induction on the rank therefore splits \(L\) as an orthogonal direct sum of such rank-two planes.
Replacing \(f\) by \(-f\) in each plane changes the displayed block to \[ \boxed{\begin{pmatrix}0&-1\\1&0\end{pmatrix}}, \] so after a suitable integral change of basis the matrix is a block diagonal sum of these standard symplectic blocks.