I apologize if this question is a little too basic for MathOverflow, but it's somewhat outside of my background and I'm frustrated that the answer doesn't seem to be explicit in the literature even though I suspect it's easily known by the experts.

What are the lowest-dimensional irreducible representations over $\mathbb{F}_2$ of the full symmetric group $S_n$ and in particular whether there exist any of dimension $\leq n - 2$ (resp. $\leq n - 3$) if $n$ is odd (resp. even)?

I know that for each $n \geq 3$ we have the $(n - 1)$-dimensional standard representation over *any* field. Moreover, if $n$ is even, we get an $(n - 2$)-dimensional representation as a quotient of this over characteristic $2$ (because then the vector with all $1$'s lies in the standard representation but is fixed by all of $S_n$). It's easy to see that these representations are irreducible. What I would specifically like to know is whether or not they are the lowest-dimensional irreducible representations of $S_n$ over $\mathbb{F}_{2}$.

(I'm curious about slightly more general questions as well, like whether or not the analogous representations to the ones I described above are the lowest-dimensional over any given positive characteristic.)