This is a very common question about SW
tanks. No, you don't have to have a skimmer. Skimmerless tanks can be successful, but are best left to experienced SW
aquarists and most of them have one laying around for times when they see the need to run it for a while. I highly recommend that you get one though. The skimmer will pull a lot of the organics out of the water before they become nitrate. My challenge to most folks who say a skimmer is not needed is this: Put a skimmer on your tank for a week and then drink what's in the collection cup...then I'll agree the skimmer isn't needed.
For your tank, my first recommendation would be an Aqua C Remora with a Maxi Jet pump (stay away from the Rios). Next would either be a Prizm or a Seaclone. There is a large price difference in these skimmers. The Aqua C will outperform the other two by a large margin which justifies the price. Whatever you get, try to obtain a surface skimmer with it...this will increase your efficiency immensely.