(eng): In this work we completely characterize generalized Young measures generated by sequences of gradients of maps in $W^{1,1}(\Omega-{R}^M)$, where $\Omega\subset{R}^N$. This characterization extends and completes previous analysis by Kristensen and Rindler [Arch. Ration. Mech. Anal., 197 (2010), pp. 539--598 and 203 (2012), pp. 693--700] where concentrations of the sequence of gradients at the boundary of $\Omega$ were excluded. As an application of our result we study the relaxation of non-quasiconvex variational problems with linear growth at infinity, and, finally, we link our characterization to Souček spaces [J. Souček, Časopis Pro Pěstování Matematiky, 97 (1972), pp. 10--46], an extension of $W^{1,1}(\Omega-{\mathbb{R}}^M)$ where gradients are considered as measures on $\bar\Omega$.