Duràn wrote down an explicit formula for such map in "Pointed Wiedersehen Metrics on Exotic Spheres and Diffeomorphisms of $S^6$". That is he wrote an explicit formula for an exotic diffeomorphism from $S^6$ to $S^6$ which is homotopic but not isotopic to the identity. This the produces an explicit homeomorphism from $S^7$ to an exotic sphere by glueing as described by Ryan in his answer.
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" by Duràn and Rigas and Sperança (this link is freely accessible unlike the first one).