We present a bracketing-based encoding that can be used to represent any 2-planar dependency tree over a sentence of length n as a sequence of n labels, hence providing almost total coverage of crossing arcs in sequence labeling parsing. First, we show that existing bracketing encodings for parsing as labeling can only handle a very mild extension of projective trees. Second, we overcome this limitation by taking into account the well-known property of 2-planarity, which is present in the vast majority of dependency syntactic structures in treebanks, i.e., the arcs of a dependency tree can be split into two planes such that arcs in a given plane do not cross. We take advantage of this property to design a method that balances the brackets and that encodes the arcs belonging to each of those planes, allowing for almost unrestricted non-projectivity ( 99.9% coverage) in sequence labeling parsing. The experiments show that our linearizations improve over the accuracy of the original bracketing encoding in highly non-projective treebanks (on average by 0.4 LAS), while achieving a similar speed. Also, they are especially suitable when PoS tags are not used as input parameters to the models.