I started noticing more bruises after being on Prozac for a while, after I had raised the dose from 20 to 40 mgs. I got really scared, fearing that I would have to stop the med (that was working wonderfully)... I called my PDoc and he did some calling around, I think he even called poison control or something, and said that it appears to happen to a minority of people.
If you do research on the web, you will find some mentioning of this too, enter ssris and bruising into a search engine, you will probably find some info... I remember somewhere it was even listed as one of the common side effects, together with the tip to avoid aspirin.
My bruising never got so bad that my whole arms were covered with bruises, but I had quite a few of them, it seemed at all times. I stopped taking aspirin for headaches and switched to tylenol, and that helped. (I tried it out one time, taking an aspirin, and immediately I had a bruise again!!!!!)
In my case I could stay on the prozac, fortunately.
I would talk to your doc, ok?