Let $q$ denote a prime power and $\text{GL}_n(q)$ and $\text{U}_n(q^2)$ the general linear and unitary group, respectively. Then $\text{U}_n(q^2)$ is naturally a subgroup of $\text{GL}_{n}(q^2)$, so on kind of groups can be embedded into the other. Does the converse hold?

My question is if there are any functions $f,g:\mathbb{N}\rightarrow \mathbb{N}$ such that we can always find an embedding of $\text{GL}_n(q)$ into $\text{U}_{f(n)}(q^{g(n)})$.

My feeling is that this is not possible, but I somehow fail to find an explanation.

A further question is what happens if we exchange the unitary groups for orthogonal or symplectic groups.