The other day i got my blood drawn from my inner elbow, right after i experience bruising on my wrist and on top of my hand,is it normal??

Bleeding: I suspect they hit a large vessel and the blood dissected down to your wrist and hand. It is possible you may have a problem with coagulation of your blood. If you notice easy bruising or bleeding you need to get checked out for your bleeding problem. The blood will be reabsorbed and should cause you no problem.

Arterial puncture?: This is unusual. Was it an arterial blood draw, e.g. for blood gases" test to check heart or lung problem? If not, I wonder about a mistaken draw from artery instead of a vein. If so, it probably isn't harmful. Still, you might want to see your doctor. If s/he agrees it might have been an arterial puncture by mistake, the person who drew your blood (or his or her supervisor) would want to know.