Brash and Maher were unable to carry out a clear round after both bringing down fences and suffering four- and eight-point penalties respectively.
Skelton, who helped win gold in the team event on Monday, looked close to edging into the medal positions until his horse, Big Star, knocked down one of the fences in the final hurdle.
Switzerland's Steve Guerdat was the only rider to complete a clear round to grab the gold medal. However, the silver and bronze medals were forced to be decided by a jump-off involving the Netherlands' Gerco Shroder and Ireland's Cian O'Connor.
Despite a tight contest, it was a clear round by Shroder, who nabbed the silver, which left O'Connor with the bronze medal.