Eskildsen studied at a school for languages (1963-67) and was apprenticed in a portrait photography studio. From 1967 to 1969 he studied photography and photohistory at the Folkwang Schule für Gestaltung in Essen, Germany.
Since 1978 Eskilden has been curator of photography at the Museum Folkwang in Essen. In 1976-77 he served an internship at IMP in Rochester. He previously was assistant curator for photography exhibitions at the Museum Folkwang, an assistant in the Photography Department at the University of Essen (1972-74) and did freelance photography for periodicals and for a publishing house (1970-72).
Eskildsen has belonged to DGPh since 1975.