In a way, that is sort of already what makes the saber engine so clever, it manages to get cryogenic oxidiser out of the atmosphere at less mass cost than it would take to lug it up there.
At the moment, it only uses this to directly pass through to the engines (not via storage).
In theory if this could be made more efficient then it could create a surplus that could be stored. However, I think you are misguided in thinking this could be practically used as a substitute to filling up the LOX tanks on-the ground.
Yes, purely theoretically (engineering not an issue): what you state would be possible.
If weight/volume wasn't an issue for the machinery needed to compress, cool, and distill the super heated and super sonic atmospheric air you bump into. Nor was weight/volume for the enormous power-plant you'd need to run all of that. And you didn't need to use up much LH to do so: yes, you could use that system to fill the [partially-]empty LOx tanks on your way up.
A) You'd be loosing an awful lot of momentum to all that air you intake.
The fastest you can spit it out the back again by reacting it with LH is about 4.4kms (which is pretty good, but if you didn't manage to gain any use of the nitrogen, the average of what you intake falls to about Mach 2.5, which means this is harming your overall dV past this speed).
B) If you had near massless power generation and complex, high power, high force processing equipment: you could optimise space travel a lot more than what you suggest.