I am reading Chapter I of David Mumford, Abelian Varieties. At the beginning of the chapter it is shown the canonical isomorphisms
$H^{2}(X,\mathbb{Z})\cong Alt^{2}(\Lambda,\mathbb{Z})$.
But I would first like to understand the structure of the $H^{2}(X,\mathbb{Z})$, that is the definition. I did not find it in Mumford, D. Also I looked in Complex abelian varietieis, Herbert and Cristine, without success.
$X$ is a compact connected Lie group of dimension $g$, topologically a product of $2g$ circles $S^1$, and $$H^\bullet (X, \mathbb{Z}) = \bigoplus_{n\ge 0} H^n(X, \mathbb{Z})$$ is the singular cohomology of $X$. It is a graded ring with a "gradedly commutative" product $$\smile\colon H^p (X,\mathbb{Z}) \times H^q (X,\mathbb{Z}) \to H^{p+q} (X,\mathbb{Z}),$$ called the cup product. Mumford uses the fact that the cup product induces an isomorphism $$\Lambda^r H^1 (X,\mathbb{Z}) \xrightarrow{\cong} H^r (X, \mathbb{Z})$$ (this is clear for a product of circles, once you know the definition of the cup product) and the first singular cohomology group is then identified as $$H^1 (X,\mathbb{Z}) \cong \operatorname{Hom} (\pi_1 (X), \mathbb{Z}).$$ Mumford's argument is rather general, and to understand it you may consult any textbook on algebraic topology.