Aske was apprenticed to John Trott, a haberdasher (dealer in raw silk) and East India Company merchant.
He is remembered primarily for the charitable foundation created from his estate, which nowadays operates two schools in Hertfordshire, Haberdashers" Aske"s Boys" School and Haberdashers" Aske"s School for Girls, and others elsewhere. Aske became a Freeman of the Worshipful Company of Haberdashers in 1643 and was elected an Alderman of the City of London in 1666. Despite marrying twice, Aske had no children and left the bulk of his sizable estate, £32,000 (equivalent to £558m in 2010, against average earnings), to his livery company for charitable purposes.
The remaining £12,000 was left to form the Haberdashers" Aske"s Foundation, of which the Company is Trustee.
The charity was incorporated by a private Acting of Parliament in 1690. An almshouse and school, Haberdashers" Aske"s Boys" School, were built on 21 acres in Hoxton by 1692 to a design by Robert Hooke.
A further 1,500 acres (6 km²) in Kent were acquired to provide an annual income of over £700. The buildings were demolished in 1824 and reconstructed in 1825 to a design by the architect, David Riddell Roper.
The almshouses were closed to enable the school to expand in 1874 to take 300 boys and 300 girls, and a second and third school were opened in Hatcham, Surrey in 1875.
Haberdashers" Aske"s School, Hoxton was relocated (Hampstead for the boys and Acton for the girls) in 1898, but both schools were reunited in 1974 at Elstree on adjoining sites. The Hatcham schools are now merged as a single state school, an Academy known as Haberdashers" Aske"s Hatcham College. Aske, originally from Yorkshire, shares his name with another Robert Aske, who was executed for treason in 1537.
The earlier Robert Aske, who died unmarried, is understood to be a collateral ancestor of Alderman Robert Aske.
In 1922, another namesake, the lawyer Robert Aske, who also hailed from Yorkshire, was created a baronet.
He directed that £20,000 was to be used to buy a piece of land within one mile of The City upon which was to be built a "hospital" (almshouses) for 20 poor members of the Company and a school for 20 sons of poor Freemen of the Company.