The 26-year-old Skinner hit 12 of her 15 orange targets with her shotgun to New Zealand's Natalie Rooney's 11.
American Corey Cogdell, a three-time Olympian, won her second bronze medal by besting Spain's Fatima Galvez in a shoot-off, after each woman shot 13 of their 15 targets for the medal.
Defending Olympic champion Jessica Rossi of Italy crashed out in the semi-finals after missing five of the 15 targets, a big disappointment for the police officer after she easily claimed gold and set a world record in London 2012.
Fourth time lucky for Germany's Engleder in 50m rifle
She finished sixth out of six finalists.
Steven Scott takes bronze for GB in double trap, dances with gun to celebrate
Anna Korakaki takes gold in shooting for Greece to bag second medal