If e and f are idempotents in a regular semigroup then the ordered pair (e,f) is said to be a skew pair if ef is not idempotent whereas fe is idempotent. There are four distinct types of skew pairs of idempotents. The regular semigroups of smallest cardinality that contain precisely one pair of each of the four types will be described as quotients of Rees matrix semigroups, as will be their occurrence in full transformation semigroups.