I'm particularly interested in the case $\Lambda^3 \mathbb{F}_3^n$, and specifically, just stabilizers of vectors that satisfy the two conditions (i) there are no zero coordinates (in the basis induced from the standard basis of $\mathbb{F}_3^n$) and (ii) they are in the image of a map $(\mathbb{R}^n)^3 \to \Lambda^3 \mathbb{F}_3^n$ that I will now describe.

We start with the obvious map $(\mathbb{R}^n)^3 \to \Lambda^3 \mathbb{R}^n$. Then write our vector in the standard coordinates $\sum_{i < j < k} a_{ijk} e_i \wedge e_j \wedge e_k \in \Lambda^3 \mathbb{R}^n$ and then replace $a_{ijk}$ with $0 \in \mathbb{F}_3$ if it is 0, $1 \in \mathbb{F}_3$ if it is positive, and $-1 \in \mathbb{F}_3$ if it is negative.

I can calculate with GAP all stabilizers for n = 4, 5, and stabilizers for given vectors for n = 6, 7.

For n = 4 I get $\mathbb{Z} / 4, \mathbb{Z} / 3$, and $Alt(4)$, for n = 5 I get $1, \mathbb{Z} / 3$ and $\mathbb{Z} / 5$, and for n = 6 and 7 I can find cyclic groups of orders 1, 2, 3, 5, 6, and 1, 3 respectively.

It seems like the sort of problem that should have a solution....