You can, Red Epic does it with what they call HDRx, Magic Lantern does it with Dual ISO, zcam E2 do it with WDR - all more or less the same things - you take two exposures for every frame. Doesn't always work for fast action, but for some shots it really helps. This is a really good blog post on the topic - https://medium.com/@jasonzhang_22759/wdr-of-e2-369a12d39a7f

Also, if your image bit depth is high enough, you can create "virtual" multiple exposures from a single image and do exposure fusion. Google already do this with HDR+ on stills.

The bit depth stored from the sensor could be a lot higher than the current 8bits most mobile phones use for video, if you have full control of the ISP. Cinema cameras can process and store 12-16bit video (Red, Blackmagic, Arri).

Even Apple does this to some degree with their "extended dynamic range" video on newer iPhones, though the saved files are still 8bit.