Skew-symmetric bicharacters on $A$ and $H^2(A,mathbb{C}^{times})$

Let $A$ be a finite abelian group. I have read that there is an isomorphism of abelian groups between the group $H^2(A;mathbb{C}^{times})$, and the group of skew-symmetric bicharacters on $A$, i.e. bilinear maps $b:Atimes Arightarrow mathbb{C}^{times}$ such that $b(x,x)=1$ for every $xin A$. How does one prove this result?

In particular, is it true that this isomorphism is not induced by the canonical map: $${bicharacters}rightarrow H^2(A;mathbb{C}^{times})$$