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.