Skelton, who takes up his role in May, has more than a decade of experience in both the public and private sectors, specialising in policy development and public service reform.
His previous roles include deputy director and head of research at the Policy Exchange think tank and principal consultant at PA Consulting.
Most recently, he founded campaign group Renewal, which aims to strengthen the appeal of the Conservative Party.
Skelton said: "I’m hugely excited about working with Weber Shandwick to help clients respond to unique public policy challenges in uncertain times."
Jon McLeod, chairman, UK corporate, financial and public affairs, said: "David’s appointment reaffirms our commitment to cross-party public affairs consultancy at a time when the political scene will remain fluid for the foreseeable future. He also brings that combination of intellectual rigour and creative thinking that is the hallmark of Weber Shandwick’s approach to public affairs consultancy."