JSFiddle - React, Tailwind, and code Playground

HTML

<div class="select-estiloso">
   <select>
      <option>Aqui a primeira opção</option>
      <option>E aqui a segunda opção</option>
   </select>
</div>

CSS

.select-estiloso { /* <div> */
       width: 240px;
       height: 34px;
       overflow: hidden;
       background: url(nova_setinha.jpg) no-repeat right #ddd; /* novo ícone para o <select> */
       border: 1px solid #ccc;
    }   

    .select-estiloso select { /* <select> */
       background: transparent; /* importante para exibir o novo ícone */
       width: 268px;
       padding: 5px;
       font-size: 16px;
       line-height: 1;
       border: 0;
       border-radius: 0;
       height: 34px;
       -webkit-appearance: none;
    }

    .select-estiloso select:hover { /* <select> */
       color: red;
    } 

    .select-estiloso select:active { /* <select> */
       color: blue;
    }  

    .select-estiloso select:focus { /* <select> */
       color: green;
    }