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1.
Looking for sex partners. Usually used to refer to a male seeking female company. But it could possibly be used to refer to a lesbian seeking sex partners or perhaps to a gay man seeking a cross-dressing or otherwise effeminate male partner.

Accusing someone of "Chasing Skirts" can imply that the accused person is usually unsuccessful in his\her\other's endeavours.
"Did you see Brad? Man, does that guy ever do anything *besides* chasing skirts?"
by Aedwynn November 02, 2005
 
2.
To go out as look for some bitches.
Mike and Doug went chasing for skirt on Friday.
by Insane V April 23, 2004