I parallel transported a vector around a parallelogram formed by another two vectors to get the Riemannian tensor:
$$R^m{}_{lkj} = \left( \partial_k \Gamma^m{}_{jl} + \Gamma^m{}_{kn} \Gamma^n{}_{jl} \right) - \left( \partial_j \Gamma^m{}_{kl} + \Gamma^m{}_{jn} \Gamma^n{}_{kl} \right) $$
The difference between the two terms is the exchange of $j$ and $k$ indexes. So the skew symmetry of the last two indices ($R^m{}_{lkj} = -R^m{}_{ljk} $) are immediately apparent.
But it isn't immediately obvious to me why there is a skew symmetry in the first two indices too, so
$$R^m{}_{lkj} = -R^l{}_{mkj} $$
How can one prove that this symmetry indeed hold?