But you really shouldn't. I'll show you two vertical videos, which will illustrate why you should shoot horizontally.
This video was shot with my iTouch. I didn't turn my device. Now the overall image is "smaller" than it would be if I had turned my iTouch. Unfortunately, those black bars are distracting. Remember that TV screens, laptop monitors, and YouTube videos are oriented to display horizontally. So your camera and cell phone screen should be turned that way, too.
I'll let Wired explain why that is:
Videos, unlike photos, are almost universally presented horizontally. There’s a reason for this: It’s how we’re built to view the world. Our vision allows us to see more to the left and right than top and bottom. So when you shoot a video on your smartphone in portrait mode, you’re violating not only the set video standard, but also the laws of nature as they pertain to human sight.Gizmodo asks, "What would it be like if every time someone gave you a dollar to spend, you spent 33 cents and threw the other 67 cents in the trash? This is like that."
Even worse, the black bars indicate that I have no idea what I'm doing. Amateur hour!
This video is even worse. Because I didn't edit the video and have it rotated, there are no black bars—but the video is unwatchable.
So, before you fire up your phone or digital camera, visualize the finished product, streaming online for your friends and family, or playing on your living room television. Then proceed accordingly, so your finished product is supremely watchable, and able to be enjoyed.
That’s Not How You Use That: Shooting Video in Portrait Mode, Wired
Portrait video and other things cameras shouldn't let you do, The Verge
PSA: Please Turn Your Damn Cellphone Sideways When Recording Video, Gizmodo