There are many ways to do it! It's a pretty natural approach if you ask me. Recall that the scalar spherical harmonics are eigenfunctions of the Laplacian on the 2-sphere. If you work through the algebra, you'll see that the grad of a Ylm is also an eigenfunction, and so is a "curl" (Hodge dual)

For way more than you wanted to know, see Thorne's 1980 review in RMP, _Multipole expansions of gravitational radiation_