I think both the man and woman should offer to pay the whole bill. I think that whoever chooses the place should pay, however I have 2 caveats.
Second, third and fourth dates etc should form a balance. One party shouldn't pay all the time, even if they are doing more of the asking.
Also, if the date has gone poorly, I think you should split the bill. To me, it's a clear signal that you won't be seeing the person again, and thus the paying won't have a chance to even itself out.