Orthogonal Lie algebra

Example

Let $k$ be a field and $n \ge 1$. The orthogonal Lie algebra is $$\mathfrak{so}_n(k) = \set{X \in \mathfrak{gl}_n(k) : X^T = -X},$$ the space of skew-symmetric matrices, with the bracket $[X, Y] = XY - YX$ of the general linear Lie algebra.

Check that it is closed under brackets: if $X^T = -X$ and $Y^T = -Y$, then $$[X, Y]^T = Y^T X^T - X^T Y^T = YX - XY = -[X, Y].$$ For $\operatorname{char} k \ne 2$, a skew-symmetric matrix has zero diagonal, and $\dim \mathfrak{so}_n(k) = n(n-1)/2$.

For $n = 3$ and $k = \R$, let $L_1, L_2, L_3$ be the matrices of the maps $v \mapsto e_i \times v$, where $e_1, e_2, e_3$ is the standard basis. They form a basis of $\mathfrak{so}_3(\R)$ with $[L_1, L_2] = L_3$, $[L_2, L_3] = L_1$, $[L_3, L_1] = L_2$. So $\mathfrak{so}_3(\R)$ is isomorphic to $\R^3$ with the cross product.