For this operation you may use a fleshing tool such as shown in Figure 1. You can get one of these from a hardware, it looks like a flattened out hoe. Saw the handle to length you need and round the corners, then, sharpen the blade as sharp as you can get it. Keep that whetstone handy for you may need to re-sharpen frequently. The sharper you keep it the better.
Some people just use the scraping tool for fleshing and scraping.
All fat must be removed otherwise the hide will not absorb the brains and you will have hard places that will rip out on you in the breaking process. These are called grease burns. A good enough job fleshing will often remove the scarf skin, leaving only the hair side to scrape. Then, let the hide dry completely in the rack. The hide is now 'preserved' in a state of rawhide, which has many uses in itself.