I know that for any finite Galois extension K of $\mathbb{Q}$ the group $H^2(Gal(K/\mathbb{Q}),K^*)$ is isomorphic to the Brauer group $Br(K/\mathbb{Q})$. The isomorphism goes as follows: to a 2-cocyle $a_{s,t}$ you associate the crossed product $(K,\mathbb{Q},a_{s,t})$ which is a central simple algebra over $\mathbb{Q}$.
I read that $H^2(Gal(\overline{\mathbb{Q}}/\mathbb{Q}),\overline{\mathbb{Q}}^*)$ is isomorphic to $Br(\mathbb{Q})$ but I can't manage to understand how the isomorphism works. My question is: for a 2-cocyle $a_{s,t}\in H^2(Gal(\overline{\mathbb{Q}}/\mathbb{Q}),\overline{\mathbb{Q}}^*)$ what is the division algebra corresponding in $Br(\mathbb{Q})$?