Or, working with cylindrical coordinates from the start, we can write the position vector of any point on the cone as .
The derivatives give vectors in the tangent plane:
The cross product of those two vectors gives the normal vector
where the order of multiplication has been chose to make the component negative ("with downward orientation").
The "vector differential of surface area" is
Of course, so that just as TheEmptySet says (with the sign changed of course).