imagine v as the sum of 2 vectors, 1 parallel and 1 perpendicular to n
then try to imagine the cross product with each one
Note that v x n is perpendicular to v and n. Hence n x (v x n) is the cross product of two orthogonal vectors, and thus has magnitude |n| |v x n| or just |v x n|.
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.