<connectionStrings>
    <add name="RHContext" providerName="System.Data.SqlClient"
           connectionString="Server=SERVER;Database=BASE;
     Trusted_Connection=false;Persist Security Info=True;User Id=USUARIO;
     Password=SENHA;MultipleActiveResultSets=True"
  </connectionStrings>

ConfigurationManager.ConnectionStrings["RHContext"].ConnectionString

public class AppConfigManager {

    //Objeto para manipular o App.config
    private static Configuration config = ConfigurationManager.OpenExeConfiguration(ConfigurationUserLevel.None);

    #region Singleton

    private static AppConfigManager instance;

    public static AppConfigManager Configuration
    {
        get
        {
            if (instance == null)
                instance = new AppConfigManager();
            return instance;
        }
    }

    private AppConfigManager()
    {
        this.ConnectionString = GetConnectionString("RHContext");
    }
    #endregion

    /// <summary>
    /// Retorna a string do App.config.
    /// </summary>
    public string ConnectionString { get; private set; }


    /// <summary>
    /// O path do arquivo de configuração de conexão
    /// </summary>
    public string ConnectionConfigPath
    {
        get
        {
            return Application.ExecutablePath + ".config";
        }
    }


    /// <summary>
    /// A string de conexão
    /// </summary>
    /// <param name="key"></param>
    /// <returns></returns>
    public string GetConnectionString(string key)
    {
        try
        {
            return ConfigurationManager.ConnectionStrings[key].ConnectionString;
        }
        catch (Exception)
        {

            throw new Exception("Não foi possivel obter a conexão do nome" + key);
        }

    }

    /// <summary>
    /// Altera a string de conexão do nome informado.
    /// </summary>
    /// <param name="connectionString">String a ser utilizada</param>
    public void ChangeConnectionString(string connectionString, string name)
    {
        //setando a string a ser alterada
        var cfg = config.ConnectionStrings.ConnectionStrings[name];

        //altera a string de conexão
        cfg.ConnectionString = connectionString;

        // Salva o que foi modificado
        config.Save(ConfigurationSaveMode.Modified);

        // Atualiza no app o bloco connectionStrings
        ConfigurationManager.RefreshSection("connectionStrings");

        // Recarrega os dados de conexão
        Properties.Settings.Default.Reload();
    }

     /// <summary>
    /// Add uma string de conexão.
    /// MeuAppConfig é a classe onde será mapeados os atributos da connectionString
    /// </summary>
    public static void AddConnectionString(MeuAppConfig app)
    {
        try
        {
            if (string.IsNullOrWhiteSpace(app.User))
                app.User = "";
            if (string.IsNullOrWhiteSpace(app.Password))
                app.Password = "";

            SqlConnectionStringBuilder scsb = new SqlConnectionStringBuilder
            {
                DataSource = app.ServerName,
                InitialCatalog = app.Database,
                UserID = app.User,
                Password = app.Password,
                IntegratedSecurity = "False"
            };

            // add a conexão
            config.ConnectionStrings.ConnectionStrings.Add(
                new ConnectionStringSettings
                {
                    Name = app.ConnectionName,
                    ConnectionString = scsb.ConnectionString,
                    ProviderName = "System.Data.SqlClient"
                });

            // salva conexão
            config.Save(ConfigurationSaveMode.Modified);
        }
        catch (Exception ex)
        {
            //Entrada já existe ou outro ...
        }
    }

    #endregion

}