  Motivated by applications to low-rank matrix completion, we give a
combinatorial characterization of the independent sets in the algebraic matroid
associated to the collection of $m\times n$ rank-2 matrices and $n\times n$
skew-symmetric rank-2 matrices. Our approach is to use tropical geometry to
reduce this to a problem about phylogenetic trees which we then solve. In
particular, we give a combinatorial description of the collections of pairwise
distances between several taxa that may be arbitrarily prescribed while still
allowing the resulting dissimilarity map to be completed to a tree metric.
