If your injury is that deep and you don't go to a doctor, you are putting yourself in serious danger. If infection set in, the wound could go gangrenous and begin spreading throughout your system. In that case, the only way to save your life would be to amputate your arm.
Please, do to the doctor. They'll probably clean the wound out and stitch it up so it can heal properly and without infection. It's not worth losing an arm because you wouldn't go to the doctor now, when the treatment would be very simple and easy.