Duràn wrote down an explicit formula for such map in ["Pointed Wiedersehen Metrics on Exotic Spheres and Diffeomorphisms of $S^6$"][1]. 
Geometric properties of that particular map were later studied by various people.
For example, it's written down explicitly on page 1 in ["Bootstrapping $Ad$-Equivariant Maps, Diffeomorphisms and Involutions"][2]  by Duràn and Rigas and Sperança (this link is freely accessible unlike the first one).


  [1]: http://www.springerlink.com/content/v775g60776g6m813/
  [2]: http://www.ime.usp.br/~eufrasio/matcont/pt/edicoes.php?idvolume=29