An antipodal set in a compact Riemannian symmetric space is a subset where the restriction of the symmetry at each point is the identity, which was introduced by Chen and Nagano. A maximal antipodal set is a kind of frame of a compact Riemannian symmetric space. In this talk I mainly treat maximal antipodal sets of oriented real Grassmann manifolds. We can reduce the classification of these to a combinatorial problem and classify these in the case where the rank is less than five. Moreover we can estimate the cardinalities of maximal antipodal sets in cases of higher rank.