  Rota's Basis Conjecture is a well known problem from matroid theory, that
states that for any collection of $n$ bases in a rank $n$ matroid, it is
possible to decompose all the elements into $n$ disjoint rainbow bases. Here an
asymptotic version of this is proved. We show that it is possible to find
$n-o(n)$ disjoint rainbow independent sets of size $n-o(n)$.
