Let \(n\geq 1\), and let \[ J \coloneqq \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix} \in \GL_{2n}(\CC). \] The symplectic group is \[ \operatorname{Sp}_{2n}(\CC) \coloneqq \theset{ A \in \GL_{2n}(\CC) \st A^tJA = J }. \]
The matrix \(J\) is skew-symmetric, \(J^t = -J\), so \(\operatorname{Sp}_{2n}(\CC)\) is the group of invertible matrices preserving the alternating bilinear form \(\omega(x,y) = x^tJy\) on \(\CC^{2n}\). Replacing \(J\) by the symmetric matrix \(\begin{bmatrix} 0 & I_n \\ I_n & 0 \end{bmatrix}\) gives instead the orthogonal group of the symmetric bilinear form \(x^t\begin{bmatrix} 0 & I_n \\ I_n & 0 \end{bmatrix}y\).