If we consider twisted Gabidulin codes proposed by Sheekey as follows: Let $n, k, s$ be positive integers such that $k<n$ and $\gcd(s, n)=1$. Let $\eta$ be a nonzero element in $\mathbb{F}_{q^n}$ satisfying $\mbox{norm}_{\frac{q^{sn}}{q^s}}(\eta)\neq (-1)^{nk}$. Then the set $$\mathcal{G}_{k,s}(\eta, h) = \left\{ \sum_{i=0}^{k-1}l_ix^{q^{si}} + \eta l_0^{q^{h}}x^{q^{sk}} \;|\; l_i \in \mathbb{F}_{q^n} \right\}$$ is an MRD code with minimum rank distance $d=n-k+1$. This code is $\mathbb{F}_{q^n}$-linear if $h=0$ and the code is $\mathbb{F}_q$-linear otherwise.
How can we obtain a generator matrix of an $\mathbb{F}_q$-linear Twisted Gabidulin codes?
Generator matrix of $\mathbb{F}_{q^n}$-linear TG codes is already known and can be found in (https://arxiv.org/pdf/1911.13059.pdf).