I am dealing with a linear system of equations that I am solving by OLS:

$$
\mathbf{y} = \mathbf{X} \mathbf{p} + \mathbf{e}
$$

Where I have $n$ samples and $k$ parameters ($\mathbf{X}$ is an $n \times k$ matrix)

I would like to work out the relationship between samples size ($n$) and the parameter uncertainties contained within their covariance matrix ($\mathbf{C\_p}$).

I have established numerically (by simulating an OLS with different $n$) that the parameter variance decreases ~exponentially with $n$, but am seeking an analytical solution. Extensive googling hasn't got me there, and neither has my sub-par knowledge of linear algebra.

Apologies if this is really basic, and thanks for the help!