We give an explicit construction, for a flat map $X åS$ of algebraic spaces, of an ideal in the $n$’th symmetric product of $X$ over $S$. Blowing up this ideal is then shown to be isomorphic to the schematic closure in the Hilbert scheme of length $n$ subschemes of the locus of $n$ distinct points. This generalizes Haiman’s corresponding result for the affine complex plane. However, our construction of the ideal is very different from that of Haiman, using the formalism of divided powers rather than representation theory. In the nonflat case we obtain a similar result by replacing the $n$’th symmetric product by the $n$’th divided power product.