For a domain \(\Omega \) in a finite-dimensional space E, we consider the metric space \(M=(\Omega ,d)\) where d is the intrinsic distance in \(\Omega \) . We obtain an isometric representation of the space \({\text{ Lip}_{0}(M)}\) as a subspace of \(L^{\infty }(\Omega ;E^{*})\) and we use this representation in order to obtain the corresponding isometric representation for the Lipschitz-free space \(\mathcal {F}({M})\) as a quotient of the space \(L^{1}(\Omega ;E)\) . We compare our result with those existent in the literature for bounded domains with Lipschitz boundary, and for convex domains, which can be then deduced as corollaries of our result.
