Mean free path of electron in air?

I’m interested in approximating the mean free path of an electron in air. I think I’m going to need to add something more into my approximation because currently I calculate $400m$ for the mean free path at atmospheric pressure. Say the mean radius of an air molecule (either $text{O}_{2}$ or $text{N}_{2}$) is about $R=0.15nm$, the approximate mean free path of the electron, $lambda$ at atmospheric pressure and room temperature is

begin{equation}
lambda approx frac{1}{nsigma}
end{equation}

where $n$ is the number density and $sigma$ is the collision cross section. The number density at atmospheric pressure ($1 text{atm} = 1.01 times 10^{5} text{Pa}$), is

begin{equation}
n = frac{N}{V} = frac{P}{k_{b}T} = 2.45 times 10^{25} text{m}^{-3}
end{equation}

The collision cross-section is

begin{equation}
sigma = pi (2r)^{2} = 10 times 10^{-29} text{m}^{2},
end{equation}

using the classical electron radius of $r= 2.8 times 10^{-15} text{m}$. The mean free path of the electron is then $400$ m. I recognise the assumption of the radius and nature of the collisions does not make sense for interactions of charged particles. However, working out proper collision cross-section is quite hard.

What even is a reasonable mean free path of an electron in air at atmospheric pressure, and is there any smart way to approximate it?