Fixie cog slip/skip
 

I have a fixie with flip-flop hub, I usually ride with the singlespeed side, but once I want to use in fixed gear, but when I apply back pressure on the pedal, or pulling up with the strap, like trying to skid, the cog slip or skipping some teeth, I can't describe well how it feels. The cog and the lockring was tightened up, but the problem isn't gone.
Any ideas?

The only explanation for this is that your cog and lock ring aren't tightened enough. Make sure to tighten the cog and lock ring as tight as possible

If you don't skid your undies, don't skid your bike. If you do skid your undies, learn how to wipe.

Some of those really cheap fixed gears don't take well to much use and abuse like that. I might pull it apart, grease it properly and torque it down right or have your local shop take a look.

You cog and lockring are not tight enough. If you do one good skid the way it is now, you will strip the threads off the lockring.

i would remove the cog and locking and inspect for any damage. if everything looks ok clean, grease and reinstall. next put front wheel of bike into corner, hop on and stomp down on pedal to really tighten down the cog, then tighten down the lockring. if it slips after that just ted shred or buy an e-bike.

If you don't have a chain whip to properly tighten the sprocket, try the rotafix method:

https://en.wikipedia.org/wiki/Rotafix

More likely, strip the lockring threads on the hub, which, unfortunately, is even more expensive to fix. It's also possible that the OP is experiencing this issue because the hub is already damaged. Remove the lockring and sprocket and carefully examine the threads.

IMO, skid stops should not be your default method of stopping your bike in the first place.

I stripped the sprocket threads of the rear hub of my first track bike, in 1964 (I was 13 and didn't know that you shouldn't install a British-threaded sprocket on a French-threaded hub). The mechanic in the local bike shop wrapped the hub in aluminum foil and threaded the same sprocket back on. Worked fine.

I've had that story in my holster for a long, long time, waiting for a thread like this to pop up.

Tighten the cog all the way down, then the lockring after

Not all the cogs are same
 

I once had the same issue after changing the cog. Tightening did not work. I was thinking to buy a new hub. Then i remembered there was a thin shim between the cog and the lockring. Solved it in a few minutes.