This paper constructs, for the first time, all the simple modules for the symmetric groups over an arbitrary field \(F\). For each Young diagram \(D\), the corresponding permutation module is denoted by \(V_D\). A trivial lemma shows that for every submodule \(U\) of \(V_D\), either \(U\supseteq E_D\), or \(U\subseteq E_D^\perp\), where \(E_D\) is (isomorphic to) the Specht submodule of \(V_D\). Therefore, \(E_D/(E_D\cap E_D^\perp)\) is zero or irreducible. If \(\text{char\,}F=0\), \(E_D\cap E_D^\perp\) is zero, hence the modules \(E_D\) give all the ordinary irreducible representations of \(S_n\). If \(\text{char\,}F=p\), \(E_D^\perp\supseteq E_D\) if and only if \(D\) is \(p\)-singular, so the modules \(E_D/(E_D\cap E_D^\perp)\) with \(D\) \(p\)-regular give all the \(p\)-modular irreducible representations of \(S_n\). By considering the rank of the Gram matrix with respect to, say, the standard basis of \(E_D\), the dimensions of the modular irreducible representations can, in principle, be computed.