Rotations always happen in a plane, in n-dimensions. In 3D there's a coincidence where planes and lines can be put in exact correspondence. This is the root of

Rotations always happen in a plane, in n-dimensions. In 3D there's a coincidence where planes and lines can be put in exact correspondence. This is the root of the whole "axial" and "pseudo" vectors that plague physics -- there's a (Hodge) duality being hidden, and we're stuck with Gibbs & Heaviside's vector calculus. But a plane (bivector) will always be the natural way to define a rotation.