Hmm - that's an axe cut? Did you cut a piece of flesh actually OFF? I would have expected a straight cut. I'd say go to a doc if it's free for you.

Nevertheless I think you'll do fine without a doc. Though probably 99% of people will tell you to see one. Be sensible, and treat it gently, do NOT pour any strong disinfectant on it (apparently that kills any bacteria but also kills your cells that it touches, which makes a toxic stew of dead cells).

I recently brought a family member to the ER after a bike accident that resulted in a huge skin rash. Insanely expensive and they just cleaned it up like you could at home.

That's not to mention all the other bad experiences I've had with doctors and hospitals, that make me want to avoid them.