Nineteen people survived when the plane swerved off the runway but some are in critical condition, said Development Minister Magdalena Alvarez.It is thought the left engine caught fire as the Spanair flight took off for the Canary Islands at about 1420 (1220 GMT) with 172 people on board.
Helicopters dumped water on the burning plane and some 70 ambulances attended.
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Spain's development minister told a news conference that investigators had ruled out foul play and considered the crash an accident.
As night fell, recovery teams were still removing bodies from the crash site, which is now silent and in darkness, except for the occasional flashing blue lights of emergency vehicles, the BBC's Steve Kingstone in Madrid says.
Spanish Prime Minister Jose Luis Zapatero cut short his holiday in the south of the country to visit the scene of the crash.
Speaking at the airport, he said: "The government will make all its efforts to support the families in this hard moment as they have received the news of such tragedy."
Spanish journalist Manuel Moleno, who was near the area when the accident happened, said the plane appeared to have "crashed into pieces".
"We heard a big crash. So we stopped and we saw a lot of smoke," he said.
A British man who witnessed the aftermath of the accident, Alan Gemmell, : "The whole back end of the plane was just burned to a crisp."
Safety record
Spanair flight JK 5022, bound for Las Palmas in the Canary Islands, came down during or shortly after take-off from Terminal Four at Barajas.
TV footage showed the plane had come to rest in a field.
Now the crash site is essentially a huge patch of scorched earth, our correspondent says.
Spanair's parent company, Scandinavian firm SAS, said the accident happened at 1423 local time.According to Spain's airport authority, Aena, the plane had been due to take off at 1300 but its departure had been delayed.
No details of the nationalities of the passengers on board have yet been released.
A local emergency service official, Ervigio Corral, told Efe news agency that there were many children among the victims.
The plane was a codeshare flight with German airline Lufthansa, which said it was investigating whether German passengers were on board.
The aircraft was a MD82, a plane commonly used on short trips around Europe, aviation expert Chris Yates told the BBC.
He said Spanair had a very good safety record.
Reports say it was the first crash at Barajas airport, some 13km (8 miles) from central Madrid, since 1983.
