Given
$$
A \to g A g^{-1} -  i dg g^{-1}
$$
where $A$ is the gauge field one-form $A = A_{\alpha \mu} t^\alpha dx^\mu$,  thus,
$$
dA\\ 
\to d(g A g^{-1} -  i dg g^{-1}) \\
= dg \wedge A g^{-1} + g dA g^{-1} - g A \wedge dg^{-1} + i dg \wedge dg^{-1} \\
= g dA g^{-1} + (dg \wedge A g^{-1} - g A \wedge dg^{-1} + i dg \wedge dg^{-1})
$$
where $d$ is exterior derivative. 

 - For Abelian case, the part in $(\dots)$ drops out, therefore $F=dA$
   is covariant/invariant (meaning $F$ transforms as $F \to g F g^{-1} = F$ ) as in Electromagnetism.  
 - For non-Abelian case, the part    in $(\dots)$ is non-zero, therefore
   you need the extra $A\wedge A$ as    in the non-Abelian $F = dA +
   A\wedge A$ to make $F$ covariant. This is THE major conclusion of Yang and Mills in their 1959 paper.