This article presents new full-wave analytical formulas for the partial element involving the free-space Green's function in the modeling of thin-wire structures for lightning transient analysis. In the framework of the partial element equivalent circuit (PEEC) formulation for thin conductors, the partial elements are computed by double-folded line integrals. In layered media, the interaction integrals involve the dyadic Green's function, which contains the free-space Green's function term. The direct numerical computation of these integrals is time-consuming, which hinders the practical application of the full-wave method. Therefore, quasi-static (QS) analytical formulas for orthogonal meshes are widely used, which significantly reduces the computational overhead. However, these existing analytical formulas are derived under the QS hypothesis, which is not suitable for full-wave wideband analysis. The accurate analytical calculation of these partial elements has not been investigated in a comprehensive way so far. Accordingly, this article presents new full-wave analytical formulas for the evaluation of the partial elements for a thin-wire structure. These formulas are derived based on the Taylor series expansion of the full-wave free-space Green's function. The range of applicability of the proposed formulas is analyzed in terms of frequency, length of the conductor, and distance between these conductors. Numerical results computed by the PEEC method are presented, showing that the proposed formulas outperform the conventional QS ones at high frequencies, which makes them more appropriate for wideband modeling.