To make a You Tube Video full width and preserve the height (and keep it responsive) you can use the following setup.
.videoWrapper {
position: relative;
padding-bottom: 56.25%;
/* 16:9 */
padding-top: 25px;
height: 0;
}
.videoWrapper iframe {
position: absolute;
top: 0;
left: 0;
width: 100%;
height: 100%;
}<div class="videoWrapper">
<!-- Copy & Pasted from YouTube -->
<iframe width="560" height="349" src="https://www.youtube.com/embed/n_dZNLr2cME?rel=0&hd=1" frameborder="0" allowfullscreen></iframe>
</div>