The bodies of the other victims have been taken to Moscow, where relatives have gathered to begin the task of indentifying the remains.
Earlier tens of thousands of Poles gathered at the presidential palace in Warsaw to pay their respects as sirens and church bells rang out at noon in tribute.
Others flocked to churches across the nation to lay flowers, light candles, sing hymns and pray.
A week of national mourning has been declared in Poland following the disaster.
A significant part of Poland's political establishment was wiped out as all passengers on board the plane, including senior government officials and parliamentarians, were killed on Saturday.
The heads of Poland's armed forces, the central bank governor, deputy ministers and 15 MPs were among those killed when the jet tried to land in heavy fog and crashed in a forest.
Donald Tusk, the Polish prime minister, described the accident as "the most tragic event of the country's post-war history", before flying to the crash site where he and Putin, met and laid flowers together.
Wreckage, including the engines, was scattered across a forest and parts of it burned for more than an hour.
The officials had been on their way to the city of Smolensk to take part in reconciliatory ceremonies commemorating the 70th anniversary of the Katyn massacre where Russian forces killed more than 20,000 Polish prisoners of war.
Kaczynski's wife, Polish church leaders and families of Katyn massacre victims were also killed.
Al Jazeera's Jonah Hull, reporting from central Warsaw, said there was a sombre atmosphere on the streets.
"It is really striking that so many people are gathering here, tens of thousands of them, and there is hardly a sound.
"They are hardly talking to each other. Many people have held a vigil through the night and many of them are visibly over come by shock and grief".
Following the constitution, Bronislaw Komorowski, the speaker of the lower house of parliament, has taken over as interim head of state. A presidential election would be held before the end of June.
Komorowski said he would announce the date of the poll after talks with all political parties.
Poland had been due to hold a presidential election in October, when Kaczynski was likely to have run against the liberal Komorowski.
Dmitry Medvedev, the Russian president, has appointed Putin to chair a special commission to investigate the crash.
Alexander Alyoshin, the first deputy chief of the Russian air force's general staff, said the plane's pilot repeatedly ignored instructions from air traffic controllers.
Poland declared a week of national mourning following the president's death
"The head of the air traffic control group gave a command to the crew to put the aircraft into the horizontal position and when the crew did not implement this order, several times gave orders to divert to an alternative airport," he was quoted as saying by Interfax news agency.
"Despite this, the crew continued the descent. Unfortunately this ended in tragedy."
Authorities have found both flight recorders, commonly known as "black boxes", from the jet.
The conservative Kaczynski, who had served as president of Poland since 2005, had a reputation for being incorruptible and was a popular figure.
Marek Matraszek, a political consultant in Warsaw, told Al Jazeera that politically, Kaczynski had been losing in popularity recently.
"But even his deepest enemies would not deny that he was hugely respected by the Polish people," he said.
"Many of his political opponents, while disagreeing with him politically, respected him for his career, his personality, his principles ... This will very much go forward into cementing how Poles will remember him: not as a politician but rather as a man of deep principle."
Matraszek said the loss of so many politicians would have a significant effect on the political scene in Poland.
"This is an issue that cuts across political barriers ... Every political party and every part of the political establishment has been affected. These were very senior people with a great deal of experience who will be very difficult to replace ... Many of the people who died had no real successors."