Thank you for asking your query.
There is warmth, swelling and rash over your right hand and you deny using any new soaps, cleansers or detergents. There are two possibilities here :
1. A subcutaneous infection - cellulitis. Absence of itching and warmth over the rash suggest a possibility of infection under the skin. This can happen without any wound or external injury in certain cases.
2.Irritant dermatitis. Although there is no history of any allergy, there is always a possibility of irritant dermatitis. Sometimes you might not have noticed the allergen at all. Or the allergen itself might be invisible.
The infection is most probable cause here. If the rash is spreading or not subsiding within 2 days, I recommend you to visit a Physician / Dermatologist in person. Meanwhile avoid applying any medication locally. You may apply slight warmth over the area once a day, if there is pain. Over the counter pain medications are ok to take.
Hope this helps.