How to style the with only CSS?

EDIT 2015 May

Disclaimer: I’ve taken the snippet from the answer linked below:

Important Update!

In addition to WebKit, as of Firefox 35 we’ll be able to use the appearance property:

Using -moz-appearance with the none value on a combobox now remove the
dropdown button

So now in order to hide the default styling, it’s as easy as adding the following rules on our select element:

select {
   -webkit-appearance: none;
   -moz-appearance: none;
   appearance: none;
}

For IE 11 support, you can use [::-ms-expand][15].

select::-ms-expand { /* for IE 11 */
    display: none;
}

Old Answer

Unfortunately what you ask is not possible by using pure CSS. However, here is something similar that you can choose as a work around. Check the live code below.

div { 
  margin: 10px;
  padding: 10px; 
  border: 2px solid purple; 
  width: 200px;
  -webkit-border-radius: 5px;
  -moz-border-radius: 5px;
  border-radius: 5px;
}
div > ul { display: none; }
div:hover > ul {display: block; background: #f9f9f9; border-top: 1px solid purple;}
div:hover > ul > li { padding: 5px; border-bottom: 1px solid #4f4f4f;}
div:hover > ul > li:hover { background: white;}
div:hover > ul > li:hover > a { color: red; }

<div>
  Select
  <ul>
    <li><a href="#">Item 1</a></li>
    <li><a href="#">Item 2</a></li>
    <li><a href="#">Item 3</a></li>
  </ul>
</div>

EDIT

Here is the question that you asked some time ago.
How to style a <select> dropdown with CSS only without JavaScript?
As it tells there, only in Chrome and to some extent in Firefox you can achieve what you want. Otherwise, unfortunately, there is no cross browser pure CSS solution for styling a select.