If $U$ is a smooth algebraic variety, then one can give a simple description of the lowest weight part of its cohomology: if $X$ is a smooth compactification and $j \colon U \to X$ the inclusion, then $$ W_k H^k(U,\mathbf Q) = \mathrm{Im} \left( j^\ast \colon H^k(X,\mathbf Q) \to H^k(U,\mathbf Q)\right). $$

Is there a similar direct description of the second lowest weight part, i.e. $$ \mathrm{Gr}^W_{k+1} H^k(U,\mathbf Q)?$$