A nice account of the case

$$
n =2
$$

is given by I. Reiner in his review of a paper of P.M. Cohn below

The review is very detailed, Hope the tex may compile...

(that NOT worked !)

I give then just the review to try in MR:

MR0207856 (34 #7670)
Cohn, P. M.
On the structure of the ${\rm GL}_{2}$ of a ring.
Inst. Hautes Études Sci. Publ. Math. No. 30 1966 5–53.
20.70 (16.48)

and the beginning of the review:

This well-written article encompasses a wealth of information about general linear groups over certain classes of rings.
The author generalizes many earlier results about such groups, and gives a number of new and striking results.
We proceed to describe some of the main theorems. Assume throughout that the underlying ring $R$
has a unity element and is associative, though not necessarily commutative. Denote by $U(R)$ its groups of units.
(1) Let $\text{GL}_n(R)$ be the group of $n\times n$ invertible matrices over $R$,
and $D_n(R)$ its subgroup of diagonal matrices. Let $E_n(R)$ be the group generated by the set of transvections
$\{I+ae_{ij}\colon a\in R,1\leq i,j\leq n,i\neq j\}$, where $\{e_{ij}\}$ is a set of matrix units.
Define $\text{GE}_n(R)=D_n(R)\cdot E_n(R)$, the subgroup of $\text{GL}_n(R)$ generated by elementary matrices.
Of course, $E_n(R)\Delta\text{GE}_n(R)$. The author calls $R$ a generalized Euclidean ring (GE-ring)
if $\text{GL}_n(R)=\text{GE}_n(R)$ for all $n$.

$$
\dots
$$