What are the best antibiotics for a mosquito bite which developed into small area of red swollen tender skin??

Not necessary: This is likely local inflammatory response and not infection. Give it some time and see how it looks.

The question: is do you need an antibiotics or topical anti-inflammatory or antihistamine. Antibiotics may be indicated only in the presence of secondary infection. Skin infections may respond to many different antibiotics especially the Penicillins, 1st& 2nd generation cephalosporins, macrolides, teteracycline, Clindamycins............