  We analyze the relationship between $n$-dimensional conformal metrics and a
certain class of partial differential equations (PDEs) that are in duality with
the eikonal equation. In particular, we extend the Null Surface Formulation of
General Relativity to higher dimensions and give explicit expressions for the
components of the metric and the generalized W\"{u}nschmann-like metricity
conditions. We also compute the equation that the conformal factor must satisfy
in order the metric be a solution of the Einstein equations.
