Activists say a village school was targeted in rebel-held Idlib.
The UN also warned that the coming winter could be the worst yet in Syria's five-year-long civil war.
Its Syria humanitarian effort head, Jan Egeland, said the brutal conflict had become more ruthless.
He said it was affecting increasing numbers of civilians.
Unicef said that, including the one in Idlib, five Syrian schools had been targeted since 11 October in "simply inhuman" attacks.
Emergency responders in Syria and a UK-based monitoring group said on Thursday that the death toll from the Idlib attack had increased to 35, and most of the victims were children.
It appears the bomb fell in the village of Hass, about 75km (46 miles) south-west of Aleppo. as children were getting ready to go home early because of air strikes.
The BBC has not been able to independently verify the details of the attack.
Russia has denied that either it or its Syrian allies were responsible.
A spokesman for Russia's defence ministry, Maj Gen Igor Konashenkov, said the claim was a fabrication and that a Russian drone had found the school's roof still intact on Thursday.
But the UK-based Syrian Observatory for Human Rights, which gathers information from a network of people within the country, said "warplanes - either Russian or Syrian" had carried out six air strikes on the village.
Activists shared photographs of dead bodies - many of them children - on the floor of a makeshift treatment centre.
Mr Ban said: "If such horrific acts persist despite global outrage, it is largely because their authors, whether in corridors of power or in insurgent redoubts, do not fear justice.
"They must be proved wrong."
The head of Unicef said that if deliberate, the attack had been a war crime.
He said: "This latest atrocity may be the deadliest attack on a school since the war began more than five years ago.
"Children lost forever to their families, teachers lost forever to their students, one more scar on Syria's future - when will the world's revulsion at such barbarity be matched by insistence that this must stop?"
Idlib is one of the last strongholds of the Syrian opposition, and the province has been repeatedly bombed by the Syrian and Russian air forces.
The US-led coalition against so-called Islamic State (IS) has also targeted rival jihadist fighters linked to al-Qaeda who operate there.
As part of efforts by the UN, the people of eastern Aleppo are being offered safe passage to the province, the BBC's James Longman in Beirut says.
But as these latest attacks demonstrate, he adds, nowhere in rebel-held Syria is free from violence.