If you're comfortable with not providing support for IE versions less than 9, you can experiment with using CSS transforms:
Start by enclosing your video in a container with a specified width and height:
<div id="videoWrapper" style="width: 800px; height: 240px;">
<video id="videoFill">
<source src="http://www.w3schools.com/tags/movie.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
</div>
Next, you can use JavaScript to resize the video accordingly (after verifying the client's browser type and version):
var wrap = document.getElementById("videoWrapper");
var vid = document.getElementById("videoFill");
var newScaleX = parseInt(wrap.style.width)/vid.offsetWidth;
var newScaleY = parseInt(wrap.style.height)/vid.offsetHeight;
var scaleString = "scale(" + newScaleX + ", " + newScaleY + ")";
vid.style.msTransformOrigin = "left top";
vid.style.msTransform = scaleString;
Keep in mind that I removed the controls attribute from the video tag. It's likely that after resizing the video, the navigation may appear awkward, so you might want to consider including a custom controls plugin using JS/jQuery or creating one yourself.
Here's a demo showcasing the solution (works only on IE 8 and above): https://jsfiddle.net/4ec1wxn8/