Hello,
These red dots can be due to viral infections or petechiael spots. Petechiae are pinpoint-sized red dots under the surface of the skin. They are common after skin trauma and viral infections. They should go away in a few days. But if they persist then please consult a physician and get other causes ruled out which include allergic reaction, autoimmune diseases, thrombocytopenia, ITP or leukemias.
My sincere advice is to get it examined from a dermatologist. It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your doctor. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.