Eleven men and nine women between the ages of 42 and 84 were killed.
Police said those killed were couples from the Swiss cantons of Zurich, Thurgau, Lucerne, Schwyz, Zug and Vaud, along with a three-member family from Austria and three crew members from Thurgau and Zurich.
The Junker JU52 HB-HOT aircraft was built in Germany in 1939 and is now a collectors' item.
Mount Piz Segnas and the Atlas, back right, are pictured in Switzerland where police say a several-seater plane crashed on the Piz Segnas mountain (file photo).
He said that officials expect the investigation of the cause to be "relatively complex".
Daniel Knecht of the Swiss Transportation Safety Investigation Board told Blick in a news conference that the vintage aircraft may have hit the ground near-vertically at a high rate of speed.
Officials can essentially rule out a collision with another aircraft or an obstacle such as a wire, Knecht said.
He also said there is no indication of foul play or that the aircraft lost parts or broke up before the crash.
The wreckage of the plane was in a basin at 2,450 metres (8,000 feet) above sea level surrounded on three sides by peaks, a Reuters witness said.
The plane had been returning from Locarno near Switzerland's southern border.
The crash occurred above the Swiss Alpine resort of Flims, an area popular with hikers and skiers. The heat makes air thinner, increasing the burden on the aircraft.
According to JU-AIR, a Ju-52 airplane, which seats 17 passengers along with two pilots, had crashed.
Junkers JU-52 planes were manufactured between the 1930s and 1950s. Its pilots are ex-military and professional pilots, all of them volunteers.
JU-Air runs sightseeing tours on its squadron of German-made vintage aircraft from Dübendorf military airfield near Zurich. It has suspended all flights until further notice.