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  "body": "\n## Definition\n\nLet $R$ be a ring and let $A = (a_{ij})$ be an $n \\times n$ [[square matrix|0JXZKX526R]] over $R$.\nThe matrix $A$ is *skew-symmetric* when $A^T = -A$, where $A^T$ is the [[transpose|09DXWYRD43]] of $A$.\nEquivalently, $a_{ji} = -a_{ij}$ for all $i$ and $j$.\n\n## Remarks\n\nIf $2$ is invertible in $R$, for example over $\\R$, then each diagonal entry of a skew-symmetric matrix is $0$.\nOver a field of characteristic not $2$, each square matrix is the sum of a [[symmetric matrix|0VR671PHE5]] and a skew-symmetric matrix: $A = \\tfrac12 (A + A^T) + \\tfrac12 (A - A^T)$.\nA real skew-symmetric matrix of odd order has determinant $0$.\nThe matrix $\\begin{pmatrix} 0 \u0026 2 \\\\ -2 \u0026 0 \\end{pmatrix}$ is skew-symmetric.\n\n## Explanations\n\n### L1 human\n\nIn a skew-symmetric matrix, the entry across the diagonal is the negative of the entry here.\nOver the real numbers this forces zeros on the diagonal, because only $0$ equals its own negative.\nThese matrices describe infinitesimal rotations.\n",
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