It will be good to get proper diagnosis done as it can lead to a no. of problems.But if the trauma is blunt and no sharp cut or injury along with severe pain in the hand,physiotherapy is a good option to get conservative treatment without any side effect.
Next Steps
You should go to a physiotherapist ,a paraffin wax bath followed by few hand muscle exercises(stretching as well as strengthening exercises)+some gripping exercises(with the ball &hand gripper) will be helpful.In case of any kind of mild or moderate tingling/numbness, electrical nerve stimulation may be required.
Health Tips
At home do a contrast bath of lukewarm water& normal water(dip the hand alternatively),Avoid hanging of hand to your side for a long time to avoid swelling,do frequent finger movements(make a fist & open),Avoid cold exposure......Take care&Get well soon!