  Recently there has been interest in pairs of Banach spaces $(E_0,E)$ in an
$o-O$ relation and with $E_0^{**}=E$. It is known that this can be done for
Lipschitz spaces on suitable metric spaces. In this paper we consider the case
of a compact subset $K$ of $\mathbb{R}^n$ with the euclidean metric, which does
not give an $o-O$ structure, but we use part of the theory concerning these
pairs to find an atomic decomposition of the predual of $Lip(K)$. In
particular, since the space $\mathfrak{M}(K)$ of finite signed measures on $K$,
when endowed with the Kantorovich-Rubinstein norm, has as dual space $Lip(K)$,
we can give an atomic decomposition for this space.
