React selecting option with object as attribute value

You can make use of index of options

class Selector extends React.Component {
  contructor(props) {
    super(props);
    this.state = { obj: null }
    this.handleChange = this.handleChange.bind(this)
  }

  handleChange(e) {
    this.setState({obj: this.props.listOption[e.target.value].obj})
  }

  render() {
    <select onChange={handleChange}>
     {this.props.listOption.map((option, index) =>
       <option key={index} value={index}>
        {option.name}
       </option>
      )}
    </select>
  }
}

Moreover, I set obj in state as null in the constructor Is there a
right way to do it?

I depends on your requirement. If you want to show at least one option as selected you can keep that instead of null