The process in which spores form outside a specialized end cell known as a basidium. Basidia are characteristic of the basidiomycete fungi (phylum Basidiomycota), and give rise to spores that each contain a haploid nucleus that is the product of meiosis. The spores are usually attached to the basidium by short spikes called sterigmata (singular: sterigma). In most basidiomycetes there are four sterigmata (and four spores) to a basidium.