  The gap probabilities at the hard and soft edges of scaled random matrix
ensembles with orthogonal symmetry are known in terms of $\tau$-functions.
Extending recent work relating to the soft edge, it is shown that these
$\tau$-functions, and their generalizations to contain a generating function
parameter, can be expressed as Fredholm determinants. These same Fredholm
determinants also occur in exact expressions for gap probabilities in scaled
random matrix ensembles with unitary and symplectic symmetry.
