The aim of this work is to construct a monomial and explicit basis for the space $M_{\mu}$ relative to the $n!$ conjecture. We succeed completely for hook-shaped partitions, i.e. $\mu=(K+1,1^L)$. We are indeed able to exhibit a basis and to verify that its cardinality is $n!$, that it is linearly independent and that it spans $M_{\mu}$. We deduce from this study an explicit and simple basis for $I_{\mu}$, the annulator ideal of $\Delta_{\mu}$. This method is also successful for giving directly a basis for the homogeneous subspace of $M_{\mu}$ consisting of elements of $0$ $x$-degree.