We know that the density of water is 1000 $kg/m^3$ and that of air is 1.225 $kg/m^3$. If I were to calculate the pressure of air at a square at sea level of area 1 $m^2$ in newtons, I'd need the height of that column of air, which means that I'd need the height of the atmosphere in that square.
Taking the value we know as 101325 Pascals at the surface, we get the weight of the air column as 10332.27 kg, or an air column of height 8434.51 meters or 8.5 km.
I know from prior knowledge that the atmosphere is actually much thicker than that - almost 300 km thick. Am I doing something wrong?