Let $H$ be a Hopf algebra over a commutative ring $R$, $A$ an $H$-extension of $B$ with structure map $\rho\colon A\to A\otimes H$, where $\otimes$ is over $R$ and $B=A^{{\rm co}\, H}=\{a\in A\colon\ \rho(a)=a\otimes 1\}$. Then $A/B$ is called an $H$-extension with a normal basis if $A\cong B\otimes H$ as a left $B$-module and a right $H$-comodule. Let $H$ be a finite Hopf algebra over $R$, $B$ a commutative $R$-algebra, $L=B\otimes H$, $L^\ast={\rm Hom}_B(L,B)$ and $A/B$ an $H$-extension. Then the authors show that $A/B$ is an $H$-Galois extension with a normal basis if and only if there exists an invertible element $v\in L^\ast\otimes L^\ast$ such that $$(1\otimes v)(H^\ast\otimes\Delta_{H^\ast})(v)=(v\otimes 1)(\Delta_{H^\ast}\otimes H^\ast)(v)$$ and ${}_{L^\ast}A\cong{}_{L^\ast}L(v)$ as a module algebra isomorphism, where $L(v)=B\otimes H(v)$, $H(v)=H$ as a left $H^\ast$-module algebra with multiplication $m(v)\colon\ H(v)\otimes H(v)\to H(v)$ defined by $h\otimes l\to\sum_i(h\ast v_{1i})(l\ast v_{2i})$, and $\ast$ is the right $H^\ast$-action on $H$. It is also shown that the group of the isomorphism classes of $H$-Galois extensions is isomorphic to $H^2(L^\ast,U)$ where $U$ is the unit functor.