You need stitches. Not all wounds can just heal by themselves. Sometimes the wound is too deep and the clotting process alone is insufficient to actually stop the bleeding. The reason your arm keep starting to bleed slowly again is because the wound is starting to clot and then opening up a little again. You have to go to A&E to get stitches. I know you don't want to and you may not want people to know hwat you've been doing, but that is the only way you can get your wound sorted. Also you need some antibiotics and a tetanus shot to make sure it doesn't get infected, which could be very serious.
I know it's not what you want to hear but you have to do it.