The British athlete managed to grab a spot on the podium after scoring a total of 693.2 following a 102.4 round in the final.
"Speechless. Can't actually believe it at the minute," said Skelhon. "I had a couple of shots that I messed up and without them I could have got to the finals in a more comfortable position.
"I thought I was going OK because the crowd's cheers were quite loud, so I looked down then and suddenly I was third."
UA Emirates athlete Abdulla Alaryani secured the gold medal.