The rash you reported after hitting your arm might be inflammation of the skin or the tissue beneath it. It might be manifested with redness, hot and swollen skin. This inflammation might not be related to any allergic reaction and, therefore, might not be called a true rash. However, it would be very important to be seen by a doctor in order to prevent further spreading of the inflammation. If the inflammation spreads to the connective tissue beneath the skin, it might turn to cellulitis.

The lack of visible entrance for the bacteria does not mean that bacteria did not enter the tissue and cause inflammation. It only means that the entrance is so small that it is not visible with the eye. Both the present condition and the possible cellulitis might be treated with antibiotics applied topically or taken orally under doctor’s supervision after examination either by your family doctor or a dermatologist.