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.
Rio 2016
Fourth time lucky for Germany's Engleder in 50m rifle
11/08/2016 AT 16:36
She finished sixth out of six finalists.
Rio 2016
Steven Scott takes bronze for GB in double trap, dances with gun to celebrate
10/08/2016 AT 19:03
Rio 2016
Anna Korakaki takes gold in shooting for Greece to bag second medal
09/08/2016 AT 20:43