Is it possible for me to be allergic to mosquitoes?

Possible: One could be allergic to mosquitos but is one does not have any symptoms of allergic reaction after a bite then clinically one is not allergic.

Yes: 1 type of localized allergic reaction to a mosquito bite is called Skeeter Syndrome. Reaction to polypeptides in the mosquito's saliva can cause inflammation (red, swollen, tender) & swelling. A normal response to mosquito bite is development of a soft white bump that later transitions into a hard, brown (reddish-brown) or a blister or bruising. There may be itching & swelling around the lesion.

Most do: Most people developed allergic sensitization after prior mosquito bites accounting for part but not all of the swelling and itching from future bites. Do note that the venom itself contains chemicals which can induce redness, swelling, and itching. Rarely, even anaphylactic reaction can occur.