JSFiddle - React, Tailwind, and code Playground

HTML

<div class="styled-select">
    <select id="billing_country" name="billing_country" size="1">
        <option value="Česká republika">Česká republika</option>
        <option value="Slovenská republika">Slovenská republika</option>
    </select>
</div>

CSS

select
{
  width:50%;
  height:38px;
  background: #FFF;
  -moz-box-shadow: inset 0 1px 1px rgba(0,0,0,0.1);
  -webkit-box-shadow: inset 0 1px 1px rgba(0,0,0,0.1);
  box-shadow: inset 0 1px 1px rgba(0,0,0,0.1);
  border: 1px solid #d2d2d2;
  color: #888;
  margin-bottom:30px;
  font-size: 12px;
  -webkit-border-radius: 5px; 
  -moz-border-radius: 5px;
  border-radius: 5px;
  padding: 10px;
  overflow: hidden;
}

.styled-select select {
  background: #FFF;
  -moz-box-shadow: inset 0 1px 1px rgba(0,0,0,0.1);
  -webkit-box-shadow: inset 0 1px 1px rgba(0,0,0,0.1);
  box-shadow: inset 0 1px 1px rgba(0,0,0,0.1);
  border: 1px solid #d2d2d2;
  color: #888;
  font-size: 12px;
  -webkit-border-radius: 5px; 
  -moz-border-radius: 5px;
  border-radius: 5px;
   background: transparent;
   width: 268px;
   padding: 5px;
   line-height: 1;
   height: 38px;
   -webkit-appearance: none;
}

.styled-select {
   width: 240px;
   height: 38px;
   overflow: hidden;
   background: url(http://cdn.bavotasan.com/wp-content/uploads/2011/05/down_arrow_select.jpg) no-repeat 205px #fff;
    background-size: 25px 25px;
   border-right: 1px solid #ccc;
  -webkit-border-radius: 5px; 
  -moz-border-radius: 5px;
  border-radius: 5px;
}