How to implement singleton design patter for database connection in C# language ?

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C#3.5

Hi everybody,
i have winform c# visual studio 2012
i want to implement "singleton design pattern"
in fact i don't want to open a new connection every time i need to request the database.
before going to the database, i want to check if the connection is already open or not
if it opens, so no need to create a new instance
otherwise, create.
And in every page of my app i just want to call my singleton whenever i need data from database without having to open or close again a DB connection

What I have tried:

namespace GNMS
{
    public class Godaddy
    {
        //singleton has a reference to itself
        private static Godaddy dbconec;
        private Godaddy() { }//private constructor so that it cannot be instantiated outside this class

        //grabs instance of singleton pattern
        public static Godaddy GetInstance()
        {
            if (dbconec == null) //check if  an instance has been created else  can create a new instance
            {
                dbconec = new Godaddy();
            }
            return dbconec;
        }

        public string connectDB() //Database connection 
        {

            try
            {

                //create a string connection
                string strconnection;
                strconnection = "Server=x.x.x.x; Database =xxx; Uid = xxx; Password =xxx;CharSet=utf8; Connect Timeout=60;";
                //initialize the connection to DB
                MySqlConnection conn = new MySqlConnection(strconnection);

                conn.Open();

                return strconnection;

                //conn.Close();

            }

            catch (Exception ex)
            {

                return ex.Message;
            }

        }

        ////
    }
}