expressing product as Vandermonde determinants

Is it possible to express the product: $$\frac{\prod_{i < j} (a_i - a_j)(b_i - b_j) }{\prod_{i,j} (a_i - b_j) }$$ as the determinant of a single matrix ?

This comes from a physics paper. Should be similar to a Vandermonde determinant.

EDIT: Obviously, I do not want a $1 \times 1$ matrix whose single entry as the answer. Why are Hilbert matrices or Toeplitz matrices or Cauchy matrices, the natural choices then ? Sorry.

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note that this still doesn't fix it, since I could still construct a diagonal matrix with say $a_1 - a_2$ as the first entry, $a_2 - a_3$ as the second entry, and the rest of the product as the third entry, and the rest of the entries as 1. Of course, we could also have a diagonal matrix where the entires are these 'polynomials'. –  Calvin Lin Dec 29 '12 at 23:50

You might want to check out Hilbert matrices, which have entries $\frac {1}{a_i + b_j}$ and determinant $\frac { \prod (a_i - a_j)(b_i-b_j)}{\prod (a_i + b_j) }$.
Replace all the $b_i$ by $-b_i$ in the Hilbert matrix and you'll get exactly what the OP wants (up to $\pm 1$). –  David Speyer Dec 29 '12 at 18:10