Affine equivalence of hyperplanes

Question

I am studying the affine equivalence of conics and quadrics (in $R^2$ and $R^3$), and I've read that all hyperplanes are affinely equivalent (I think the statement is generally referred to $R^n$ space). Can I prove this by simply saying that if I take $n$ points on one hyperplane and $n$ on the other,it always exists an affinity which sends the firsts to the seconds? If not, how could I prove this fact?