The basic unit of Pascals, (Pa), is one Newton (N) per meter square (area). Why is this, especially for things like air pressure that deal with three dimensional space? Shouldn't something like air pressure (e.g. atmospheric pressure) be in newtons per unit volume? (e.g. N/m^3)
Air pressure itself doesn't act on a three-dimensional space! It's the pressure on a surface exerted by the air. For example, imagine pressurizing a bike tire. The air inside the tire can only push against the inner surface of the tire, which is two-dimensional.