Meh, you're right: the dimension of the space of 2-dimensional subspaces of n-space is 2n-4, not 2n-3. The reason why my handwavy dimension-counting above was wrong is ("of course") that I failed to "subtract one because of the equivalence class of rotations". And yes, you're right that in general it's k(n-k).

"Dimension" here means: locally the set looks like a that-many-dimensional vector space. That is, e.g., any element of SO(n) has a neighbourhood that's topologically the same as a neighbourhood in R^(n(n-1)/2).