sorta, kinda. Of course the police have to hold it as "evidence" until the trial. I once witnessed a car crash right next to my truck in Burbank, and the perp then tried to drive off (one of his axles was broken, so that didn't work too well). But as he turned to drive over the median, I got a perfect view of the license plate and wrote it down on my mileage log. The cops who arrived demanded that I turn over my log to them (they offered to make a photocopy if I went to the main police station in three hours). I asked the officer what would have happened had I written the plate number on my hand. He just glared at me.

Had the guy committed suicide in your car - you would have had a genuine mess to deal with. The cops throw a white cloth over the vehicle and tow it off, and insurance would total the car as biohazard.