After graduating as an architect from the Fine Arts Academy in Brussels, he took refuge during the First World War in the presbytery of Nuenen, where Bart de Ligt was a pastor.
Up until the end the war, Eekman exhibited his work frequently in Dutch museums and collectors purchased some of his artwork, primarily Helene Kröller-Müller.
There he became friends with Jean Lurçat, Louis Marcoussis, André Lhote, Marc Chagall, Picasso, Dalí, Fernand Léger and Max Ernst, among others.
In 1937, at the International Exhibition in Paris, Eekman won the gold medal for his painting La pelote bleue, which was purchased by the State for the Jeu de Paume Museum.
In 1961, an important fresco (2.50 x 1.40 m. – 8.2 x 4.5 ft.) of medicinal plants from around the world was commissioned to Eekman by Prof. H. Griffon and was completed for Paris Orly Airport's drugstore.