Dean Lupton, 28, from Annan, was found between neighbouring Annan Academy and Hecklegirth Primary in the Dumfries and Galloway town yesterday morning.
The discovery was made by a member of the public at around 5.45am.
The schools were shut for the day following the discovery and were scheduled to reopen today.
Police said there do not appear to be any suspicious circumstances.
A report will be submitted to the Procurator Fiscal.