Yes, Skunks are perfectly capable of climbing.
The Striped Skunk, with its long claws, will not climb trees, except when young. As they age, the long claws become a disadvantage when coming back down trees, but they are known to climb wire fencing. The Spotted Skunk is a good climber and will climb trees. It can come back down the tree head first.
Skunks are great to have in your garden for pest control and some keen gardeners in the US will attract skunks for that very purpose. Since they are nocturnal, they don't usually cause a problem, but disturbing their nest or their night time hunt can result in a most unpleasant aroma, as you and your dog have been unfortunate to discover the hard way.