Given any $k$-flat, how can you generalize a unit normal vector $\hat{n}$? Would it be that given the flat in dimension $ k$, $\hat{n}_k > 0$? What if the flat had a rotation so that $\hat{n}_k = 0$, giving two solutions?
As a recap, how do you generalize any unit normal vector normal to a flat?
Given a $k$-flat (also called a $k$-dimensional affine subspace) of $\Bbb{R}^n$, the perpendicular subspace is $(n - k)$-dimensional. The set of all unit normal vectors to the $k$-flat forms an $(n - k - 1)$-dimension sphere. (The set of unit vectors in an $m$-dimensional space is always a sphere of dimension of $m - 1$.)
In the special case $k = n - 1$, the $k$-flats are called hyperplanes and the perpendicular space is $1$-dimensional (a line); hence the set of unit normal vectors forms a $0$-sphere (two points), which is why you have a choice of two possible normal directions.
Once $n - k > 1$, the perpendicular space has more than $1$ dimension, and so there is a continuum of possible unit normal vectors. For example, when $k = n - 2$, there is a $2$-dimensional perpendicular space; hence the set of unit normal vectors forms a $1$-sphere (circle). Think about a line in $\Bbb{R}^3$. Every point has a "circle's worth" of perpendicular directions in the ambient space.