The answer to the question is yes, though I don't have all the old literature at my fingertips. This kind of question for various classes of linear groups has a long history in the study of homomorphisms and isomorphisms of classical groups and then other algebraic groups (van der Waerden, Dieudonne, ...) The most comprehensive treatment was given by Borel and Tits in their Ann. of Math 97 (1973) paper, but emphasizing simple types rather than general reductive groups. Anyway, for general linear groups the ideas occur much earlier and also involve the uniqueness of $n$. (As you point out, the case $n=1$ has a different flavor.) I'll check the sources, but you could also work back from the references in Borel-Tits.
P.S. Note that any isomorphism (of abstract groups) between two general linear groups induces an isomorphism of the derived groups. Given $n>1$, these are special linear groups and fit well into the older or newer sources I mentioned. (Probably there is enough detail in the 1928 Hamburg paper by Schreier and van der Waerden to settle your question, but I confess I've never gone back that far in the literature.)
ADDED: One relatively modern reference I should point out is Lectures on Linear Groups by O.T. O'Meara, CBMS 22, Amer. Math. Soc., 1974. While O'Meara's own research interest at the time was in the direction of linear groups over various rings of interest, these lecture notes also incorporate older work over fields. See in particular his Sections 5.5-5.6 for theorems most relevant to the question asked here.
Roughly speaking, the central concern in these isomorphism theorems is what happens to unipotent elements (classically, transvections and the like). In the setting of classical matrix groups, the original techniques rely on the underlying geometry of the situation. But in the broader treatment by Borel-Tits the structure theory of reductive groups (Jordan decomposition, Bruhat decomposition, etc.) plays the most prominent role. For general or special linear groups, it's hard for me to judge what approach is really "simplest".