Your map is *not* onto for $n>2$, even locally. If $(A,B,C)$ is a triple of $2$-forms on $\mathbb{R}^n$ that can be written in the form
$$
(A,B,C) = \bigl(\mathrm{d}\alpha + \beta\wedge\gamma,
\mathrm{d}\beta + \gamma\wedge\alpha, \mathrm{d}\gamma + \alpha\wedge\beta\bigr),
$$
then, taking the exterior derivative of these equations, we find
$$
(\mathrm{d}A,\mathrm{d}B,\mathrm{d}C)
= \bigl(B\wedge\gamma-\beta\wedge C,\ C\wedge\alpha-\gamma\wedge A,\ A\wedge\beta-\alpha\wedge B\bigr).
$$
In particular, if $A$, $B$, and $C$ vanish at a point $x\in\mathbb{R}^n$ at which $(\mathrm{d}A,\mathrm{d}B,\mathrm{d}C)$ does not vanish, then the $1$-forms $\alpha$, $\beta$, and $\gamma$ cannot exist on a neighborhood of $x$. (Such examples are trivial to construct.)

Moreover, when $n>4$, this map does not contain the generic triple $(A,B,C)$ in its image.

The cases $n=3$ and $n=4$ are special, and, for suitable genericity hypotheses, one can prove surjectivity in special cases and under the right conditions, but it is, indeed, somewhat delicate.

By the way, this is not a 'strange problem'. It is known as the problem of prescribed curvature for $\mathrm{SU}(2)$ connections. For more information, you might look at my answer to this question.