  We revisit the old idea of constructing difference sets from cyclotomic
classes. Two constructions of skew Hadamard difference sets are given in the
additive groups of finite fields using unions of cyclotomic classes of order
$N=2p_1^m$, where $p_1$ is a prime and $m$ a positive integer. Our main tools
are index 2 Gauss sums, instead of cyclotomic numbers.
