I have been reading MacLane's book on Homological Algebra, and in chapter 3 section 2 exercise 1 , he asks to do the following
For abelian groups, show that a normalized function on $C \times C$ to $A$ is a factor system for extension of abelian groups if and only if it satisfies the identities
$f(c,d)+f(c+d,e)=f(c,d+e)+f(d,e)$, $f(c,d)=f(d,c)$
Where normalized functions satisfies $f(c,0)=0=g(0,d)$, $g(r,0)=0$.
I think i was able to do this $\rightarrow$, but the other one i have a doubt that well i have these functions with these properties , am i supposed from this properties choose representative functions $u$ for elements on $C$, and show that $(f,g)$ are a factor system for this representative function $u$? Thanks in advance.
Note that the function $g$ is basically useless here, since it is supposed to encode the action of the base ring $R$ for $R$-modules, but for abelian groups the action of $R=\mathbb{Z}$ is already given by the group structure.
So as you guessed if you have such a symmetric $2$-cocycle $f:C\times C\to A$, what you want to do is define an abelian group $G$ with an exact sequence $0\to A\to G\to C\to 0$ and a set-theoretic normalized section $u:C\to G$ such that $f$ is the factor system.
It is very easy, you basically don't have any choice if you think about how factor sets are defined: take $$G = A\times C$$ with the product $$(x,c)\star (y,d)=(x+y+f(c,d),c+d).$$
Then $A\to G$ is just $a\mapsto (a,0)$, $G\to C$ is $(x,c)\mapsto c$, and the section $u:C\to G$ is $u(c)=(0,c)$.
You can check that your first identity reflects associativity, and the second one commutativity of $\star$.