CSS in STREAM-tag?

I work a lot with background videos on websites rather than image headers.

Is it possible to insert CSS in the STREAM tag so that the video can be responsive and scale correctly and thus cover the entire screen and be able to zoom in on eg. a mobile device.

Only Heigh and Width but would love to have a class = “”

Also, using STREAM may not be as cost-effective or what do you think? However, it feels very good to load the movies quickly instead of an on site mp4 but it might be very expensive?