By explicit construction of the ADHM data, we prove the existence of a charge-seven instanton with icosahedral symmetry. By computing the holonomy of this instanton we obtain a Skyrme field which approximates the minimal energy charge-seven skyrmion. We also present a one-parameter family of tetrahedrally symmetric instantons whose holonomy gives a family of Skyrme fields which models a skyrmion scattering process, where seven well-separated skyrmions collide to form the icosahedrally symmetric skyrmion.