Degrees of freedom in hyperplane intuition?

Question

I can describe an $n-1$ dimensional hyperplane in $R^n$ with a point and a single $n$ dimensional vector (namely, the normal vector). Similarly, I can describe a $1$ dimensional hyperplane (a line) in $R^n$ with a point and a single vector (the vector pointing in the direction of the line). However, in order to represent an $n-2$ dimensional hyperplane in $R^n$, I need TWO normal vectors and a point. This generalizes, and to represent an $m$ dimensional hyperplane in $R^{2m}$, I need fully $m$ vectors in $R^m$ and a point. What is the intuition behind this explosion in the number of free parameters necessary to represent a hyperplane? Is this merely an artifact of the way that I'm representing them (with normal vectors or parallel vectors), or is this an intrinsic property of embedding lower dimensional spaces in higher dimensional ones?