Let the vector
w = n x (v x n)
where v is arbitrary and n is a unit vector. In what direction does w point and how is it related to v?
Next, observe that w is perpendicular to v x n. Since v x n is in itself perpendicular to both v and n, w is on the plane spanned by v and n. Since it is perpendicular to n too, it is a vector of magnitude |v x n| on the plane spanned by v and n, that is perpendicular to n. Which side it's on is
determined by the "right-hand rule" for cross products.
Note we assume v and n are not parallel, but if they are, w is just 0.