# Definition The special orthogonal group $SO(n)$ is the [[Identity Component]] of the [[Orthogonal Group|orthogonal group]] $O(n)$. More simply, it is the group of [[Orthogonal Operator|orthogonal matrices]] with [[Determinant|determinant]] $+1$. This [[Lie group]] has [[dimension]] $n(n-1)/2$. It represents [[Rotation|rotations]] in $n$ spatial dimensions. # Lie Algebra The [[Lie Algebra]] of $O(n)$, $\mathfrak{o}(n)$, is the set of $n\times n$ [[skew-symmetric matrix|antisymmetric matrices]]. Imposing the unit determinant condition we get: $ \det{e^{tX}} = e^{t \text{tr}(X)} = 1 \implies \text{tr}(X) = 0 $ However, antisymmetric matrices are already traceless, thus, we conclude that $\mathfrak{so}(n) = \mathfrak{o}(n)$. We will always refer to them as $\mathfrak{so}(n)$. This is not difficult to understand. The Lie algebra is the [[tangent space]] of the [[Lie group]] at the identity element. In the case of $O(n)$, the identity element is contained in the [[Identity Component]], i.e., $SO(n)$. This is why we have $\mathfrak{o}(n) = \mathfrak{so}(n)$