Shame this doesn't include my favorite description, a 4d vector in projective affine coordinates.

Instead of a normal and point or constant you get (x,y,z,1) . P = 0. The translation between the two is trivial. If you want a plane spanned by 3 points you just can use the generalized cross product to find P.

One advantage is that you can avoid all the special cases with 3 intersecting planes. There exists exactly 1 point that is on all 3 planes, but as this is in projective coordinates it might lie at infinity.