Cylindrical and spherical coordinate systems are three dimensional so you would have to say what two dimensional object you want to find the area of before an area element can be given. However, we can say this- any area element is Cartesian coordinates can be written as Adxdy+ Bdxdz+ Cdydz for some A, B, C, which may be functions of x, y, and z, depending on the surface.
In cylindrical coordinates, we have x= r cos(\theta), y= r sin(\theta), and z= z so that dx= cos(\theta)dr- r sin(\theta)d\theta, dy= sin(\theta)dr+ rcos(\theta)d\theta, dz= dz. From that, we can compute, remembering that the "wedge product" of differentials is skew-commutative,
dxdy= r cos^2(\theta)drd\theta- r sin^2(\theta)d\theta dr= r cos^2(\theta)drd\theta+ r sin^2(\theta)drd\theta= r dr d\theta
dxdz= cos(\theta)drdz- r sin(\theta)d\theta dz
dydz= sin(\theta)drdz+ rcos(\theta)d\theta dz
and, of course, changing the variables in A(x,y,z), B(x,y,z), C(x,y,z) to r, \theta, and z.
