The classical Selberg integral contains a power of the Vandermonde determinant. When that power is chosen to be a square, it is easy to prove Selberg's identity by interpreting it as a determinant of one-variable integrals. We give similar proofs of summation and transformation formulas for continuous and discrete elliptic Selberg integrals. In the continuous case, the same proof was given previously by Noumi. Special cases of the resulting identities have found applications in combinatorics.