In theory antimatter and matter anihilate to pure energy. Thus if the antimatter-matter anihilation would be the source of "Bing Bang" in theory no matter should have been left. Beside, the expansion of the universe doesn't "help" condensing the matter - quite the opposite, it's expanding it and eventually the universe will "die" because of that.
However, Encyclopedia Brittanica states that:
At the higher energies characteristic of particle-antiparticle collisions taking place in colliding-beam storage ring particle accelerators or in the big-bang model of the early universe, the annihilation energy is sufficient to create heavier particles and their antiparticles, such as muons and antimuons or quarks and antiquarks.
Technically it means you could have an antimatter-matter collision and create particles by that. However, that occurs only when they collide with high energy. And I don't think it's a way to somehow break the law of conservation of energy - if there's the energy of collision + the energy(mass) of two particles, then unless something "weird" happens, the resulting energy + mass should be equal.
We don't know whether the Bing Bang occur because of antimatter-matter collision, however the real issue isn't why matter condensed (that's simply explained by gravity and as I said before, the expansion of universe resulting from the Bing Bang actually works "against" this condensing) but why there is not the same amount of matter and antimatter in the universe. This question has remains open. You might want to check out the answer to this similar question which explains it nicely: https://physics.stackexchange.com/a/196177/181410