What is the number $N^d_k$ of real-valued parameters that are needed to specify a k-dimensional subspace of $\mathbb{R}^d$? And how can these parameters be interpreted?

I know: $N^d_1 = N^d_{n-1} = d - 1 = \binom{d}{1} - 1$.

The parameters can be interpreted as the d components of a vector spanning the 1-dimensional subspace minus its (arbitrary) length.

I know: $N_1^3 = N^3_2 = 2 = \binom{3}{2} - 1$.

The parameters can be interpreted as two angles or as the three components of a normal vector of the 2-dimensional subspace minus its (arbitrary) length.

I know: $N^d_2 = N^d_{d-2} = \binom{d}{2} - 1$

I believe this, because a d-dimensional rotation has $\binom{d}{2}$ degrees of freedom, one for the rotation angle, the remaining $\binom{d}{2} -1$ ones for the (d-2)-dimensional (hyper)plane of rotation which also defines a 2-dimensional hyperplane as its orthogonal complement.

Question: How do I know that $\binom{d}{2}$ is the number of degrees of freedom of a d-dimensional rotation?

How can these $\binom{d}{2}$ parameters of a rotation or the $\binom{d}{2} - 1$ parameters of a 2-dimensional hyperplane be interpreted (maybe even intuitively)?

I guess that $N^d_k$, the number of parameters that are needed to specify a k-dimensional subspace of $\mathbb{R}^d$, is given by $\binom{d}{k} -1$. How can this be shown? Only formally by mathematical induction or more directly, using e.g. the observation, that there are $\binom{d}{k}$ k-dimensional subspaces of $\mathbb{R}^d$ spanned by k of d elements of an orthonormal basis of $\mathbb{R}^d$?