Could it be an allergic reaction from mosquitoe bites that the bites turn in to big swollen welts?

Yes and no. Most people get a local allergic reaction to mosquitoes. This will lead to a large area of redness. Some people are more allergic than others, so those people get even worse rashes and reactions. There are some lucky people who are not allergic at all. So while it is an allergic reaction, there is usually nothing dangerous about it. Oral antihistamines might help.