Check your lease, which almost certainly lists a necessary notice period.
If you did provide adequate notice, you are unlikely to need to pay the additional month's rent ("unlikely" because I am not a lawyer).
If you did not provide adequate notice then you are likely required to pay something (maybe the full month's rent, maybe enough pro-rated rent to bring the "notice period" to 20 days, maybe something else). If you are required to pay something you don't really have the option of "skipping" the payment.
If you try to skip it the landlord has a variety of options which can compel you to pay, plus extra costs in time, money, and irritation for you. Some courses of action may have additional consequences for you, such as damaging your credit, or generating bad references if future landlords contact him.
It is common in the U.S. for renters to not pay their final month's rent and let the landlord keep an appropriate amount of their security deposit (provided that the numbers line up correctly, which they sometimes do). That's not really what security deposits are for, and if the landlord assesses damages to your unit which exceed your security deposit minus unpaid rent you may still have to pay additional money. And it's still you paying the amount besides. But if you have security deposit and try to skip out on the last month's rent, don't expect to see a dime of your deposit back.