Ban announced plans to set up a probe during his visit to Gaza last month after describing Israeli shelling of UN-run schools as a "moral outrage".
Israel maintained that Hamas militants were using the schools to store weapons and denied it had deliberately targeted the facilities, which were being used as shelters by Palestinian civilians during the 50-day war.
At least five UN facilities were hit during the conflict, killing scores of Palestinians including children, according to the UN refugee agency UNRWA.
The inquiry led by retired Dutch general Patrick Cammaert will "review and investigate a number of specific incidents in which death or injuries occurred at, and/or damage was done to United Nations premises," UN spokesman Farhan Haq said.
It will focus on incidents from July 8 to August 26, he added.
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A former military adviser to Ban, Cammaert also served as UN force commander in the eastern Democratic Republic of Congo and for the mission to Ethiopia and Eritrea.
Other panel members are legal expert Maria Vicien-Milburn of Argentina, who is general counsel for the UN culture agency UNESCO, American Lee O’Brien, who has worked on Middle East affairs at the UN department of political affairs, Canadian Pierre Lemelin and K.C. Reddy of India.
The UN spokesman emphasized that Ban "expects that the board will enjoy the full cooperation of all parties concerned."
Ban toured the site of damaged UN shelters during his visit last month and declared that the destruction from the war was "beyond description."
UN officials have said that Hamas rockets were found in vacant shelters and condemned those responsible for putting civilians at risk.
The UN Human Rights Council in Geneva separately has set up a commission of inquiry into the Gaza offensive, led by Canadian lawyer William Schabas.