The road is notorius for serious accidents. A bus accident in 2010 also claimed 20 lives.
Democratic Alliance leader in the province Ivan Meyer on Friday expressed the party's condolences and sympathy and wished those injured a speedy recovery.
"Our hearts and prayers are with them," he said in a statement.
"If it is determined, however, that the accident could have been avoided in any way, I will endeavour to ensure that the proper action gets taken."
Meanwhile, a major investigation was launched into the cause of the accident, provincial transport minister Robin Carlise said.
"Early reports from the surviving passengers indicated brake failure as the main reason for the crash," he said in a statement.
"The Provincial Regulatory Entity will investigate the legal status of the bus including the validity of the operating licence, and whether the Certificate of Fitness is still valid."
He said the bus was registered in the Western Cape and was owned by Atlantic City Liners.
Condolences
Earlier, government also sent its condolences to the families and friends of the 24 people killed in the crash.
"This is an unfortunate incident. Drivers are urged to be vigilant on the road and obey all traffic laws," said acting government spokesperson Phumla Williams.
The Inkatha Freedom Party also sent its condolences to the families of those who had died.
"This is a terrible tragedy. We urge people to be extra cautious when it is raining and to follow all the rules of the road," IFP spokesperson Petros Sithole said.
"The traffic cops and police should investigate whether the bus was overloaded and if so, we would expect the necessary criminal charges to be laid against the bus driver."
Western Cape traffic chief Kenny Africa said the bus was on its way to Cape Town on Friday morning, when the driver lost control of the vehicle and it veered off the N1 on the Hex River Pass, near De Doorns.
Critically injured
ER24 spokesperson Vanessa Jackson said the double-decker bus was crushed up against the side of the mountain along the pass.
Provincial health department spokesperson Faiza Steyn said two of the 24 who died were children.
Eight people were critically injured, three of them children; 14 were seriously injured; and 44 others sustained minor to moderate injuries.
Carlisle said emergency services were on the scene. The injured were moved to various hospitals and clinics around the province depending on the seriousness of their injuries. – Sapa