The suspected bomber was reportedly blown out of the plane in the incident shortly after take-off from Mogadishu.
More than 20 people have been arrested in connection with the incident, a government spokesman told BBC Somali.
Most of the passengers, including the alleged bomber, were said to have been originally booked on Turkish Airlines.
No group has said it carried out the bombing but al-Shabab, part of al-Qaeda, is based in Somalia.
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Most of the 74 passengers on board the flight to Djibouti were transferred from Turkish Airlines, according to Mohamed Ibrahim Yasin Olad, head of Daallo Airlines, which operated the Airbus 321 that was bombed.
"We were informed about the change a few hours before departure time," he told the BBC.
Mr Olad said he was told the Turkish Airlines flight was cancelled because of bad weather.
Local media had reported that Turkish Airlines had suspended its flights from Mogadishu in December following a failed al-Shabab attack on the airport.
Turkish Airlines spokesman Yahya Ustun told Reuters news agency the carrier had cancelled a flight to Mogadishu last week due to atmospheric conditions and said the company would not make any further comment.
In the video, a man in an orange hi-visibility vest is shown walking with a man in a blue shirt holding what looks like a laptop. Another man in a hat approaches them and it is alleged that the laptop is handed over.
Aviation expert David Learmont, however, is sceptical that the video is all that conclusive.
"I don't think that anybody knows for sure if this is the transaction that was behind the bombing," he told the BBC's Focus on Africa radio programme.
The explosion on the Daallo Airlines flight happened about 15 minutes after take-off, when the plane was only at around 11,000ft (3,350m) and the cabin was not yet pressurised.
It was able to make an emergency landing at Mogadishu airport.
Serbian captain Vlatko Vodopivec said at the time he and others were told the explosion was caused by a bomb.
Daallo Airlines flies regularly from its base in Dubai to Somalia and Djibouti.
Somalia is battling militant Islamist group al-Shabab that has been carrying out deadly attacks in its quest to establish an Islamic state.
Somali authorities have released a video showing a passenger being given a laptop they say contained a bomb that blew a hole in a passenger plane.