Wash the bite area with cold soapy water. Then wash the area with warm soapy water. Then wash it with hot soapy water.
I would put rubbing alcohol on the skin when I went out, to repel the mosquitos.
BTW, its the female mosquito that bites. The male mosquito doesn't have the mouth parts to do it. The mosquito uses the blood in the reproduction process. The higher in sugar your blood is the greater the chance you have of attracting the mosquito.
If you want to get rid of the mosquitos, and don't do this if endangered species depend on the mosquito or the larve for food because it works too well, get a bug wacker. I used to live by a stream with lots of stagnant water and therefore lots of mosquitos. I used the 1/3 acre size. Find a place between your home and the water source and hang the bug wacker about head high. Go to your local butcher and get 8 oz of beef liver on a styrofoam tray. Ask the butcher for extra blood and be prepared to get a funny look from the butcher. When you get home put the tray of liver (throw the plastic covering away) underneath the bug wacker. Attach it with duct tape or rubber bands, just don't use anything metal. When dusk falls, turn the bug wacker on and retreat to the house. The next morning, turn the bug wacker off and get rid of the mosquitos. (When I did this I had so many mosquitos I couldn't see the bug wacker light.) I cleaned out the bug wacker each morning for about two weeks before I was finally able to see the light in the morning. By the end of three weeks I could swim in my pool without mosquito repellant and wasn't bothered by mosquitos.(Don't hang the bug wacker within 20 feet of the house.)
The bug wacker thing was so effective, I didn't even have to turn it on the following year. If you have questions, please e-mail me.
Good luck to you and yours.