No a skunk is not in the rodent family so the skunk would not be a rodent.
Skunks are in the Mephitidae family and Mephitidae means stink which is where skunks really got their name.
Skunks actually will eat rodents such as mice and rats when they are hungry enough so although skunks are also a nuisance and can spray you they are beneficial at keeping rats away from your property.
However I think I would rather have the mice or rats than a skunk because at least a mouse or rat cannot spray me or my pets with a stinky spray that's hard to get rid of.
The closest relatives to skunks are stink badgers and also related to polecats and other members of the weasel family.