The air on most jet powered aircraft comes from its engines. This air is called bleed air. It is both used to regulate the temperature inside the cabin and to pressurize the plane. This air is heavily filtered but also very dry. Almost all aircraft run about 4% humidity during flight. The exception is the Boeing 787 which can provide about 15% humidity.
Because the air is so dry, you are going to notice dry skin, especially on longer flights. The skin on your hands might start to crack and your lips might become chapped. This is most noticeable with frequent flyers or anyone taking multiple flights in one day or traveling several days in a row.
One thing you are going to need to do is drink a lot of water. It is easy to get dehydrated on a flight. When you are sitting in a seat for a long time you might not notice how little you are drinking. Remember that you can bring empty water bottles through the security check point and fill them up in the airport.
Another thing you should do is bring skin lotion and lip balm. The lotion will have to be TSA compliant (3.4 ounces or less) but many companies make travel size toiletries. Use these often to keep your skin moisturized and don’t wait until you start to show signs of dry skin.