> Each plane is perpendicular to one axis. This is a coincidence that only happens in three dimensions
This part is confusing to me; in any number of dimensions, your orthonormal basis vectors (coordinate vectors) will by definition have tangent vector spaces (planes). There's nothing special about 3 in this regard, unless I'm misunderstanding something.
In N dimensions you do indeed have N hyperplanes of dimension N-1. But you have N*(N-1)/2 planes of dimension 2. Then one needs to explain why is rotation in general related to a 2D plane, not to an axis. Tautologically, it is so because rotation is when you oscillate between two directions (the two vectors of the 2D plane). It is not "rotating around an axis", because there is no consistent way to give one single angle of rotation around an axis if you are in 4+ dimensions.
I think they're saying the planes in particular are the tangent vector space in 3 dimensions. They're no longer the tangent vector space in 2,4,5.. dimensions, but they're still a thing there as well.
This part is confusing to me; in any number of dimensions, your orthonormal basis vectors (coordinate vectors) will by definition have tangent vector spaces (planes). There's nothing special about 3 in this regard, unless I'm misunderstanding something.