SO(n)

The set of rotations (w.r.t right-handed frame) in \(n\) dimensional Euclidean space, represented by the set: \(\{R \in \mathbb{R}^{n\times n} : R^TR=RR^T=I\ \text{and} \det(R)=1\}\). \(SO(n)\) is closed under matrix multiplication, is isometric (preserves lengths) and \(\forall \; R \in SO(n)\;: \;R^{-1} = R^T\}\).

The case \(n=3\) is used to represent orientations in 3D Euclidean space, with many applications in engineering.