Help converting this Write to operator<< overload

I have a CSkill class which is rather complex as it is recursive. That is:

class CSkill
{
public:
    CSkill( std::string Name, float Value ): Name_( Name ), Value_( Value )
{};
    void Update( const std::string& Name, const float Value );
    float Value( const std::string& Name ) const;
    float Sum( const std::string& Name ) const;
    void Write( std::ostream& os, const std::string& Prefix ) const;

    std::string CSkill::PlainName( const std::string& Name ) const;

    friend std::istream& operator>>( std::istream& is, CSkill& Skill);

private:
    float Sum() const;

    std::string Name_;
    float Value_;
    std::map< std::string, CSkill > Skills_;

};

Well, to output this class to an ostream, I use this function:

void CSkill::Write( std::ostream& os, const std::string& Prefix ) const
{
    if ( Name_.length() > 0 && Value_ > 0 )
        os << Prefix << Name_ << " " << Value_ << std::endl;
    for ( std::map< std::string, CSkill >::const_iterator it =
Skills_.begin(); it != Skills_.end(); ++it )
    {
        (*it).second.Write( os, ( Name_.length() > 0 ? Prefix + Name_ + "|"
: "" ) );
    }
}

but I would prefer to use
std::ostream& operator<<( /* what goes here? */ )

The problem I see is that I need to pass that extra parameter, a
std::string, which is used in the recursion. Is something like this
allowed?
outfile << Skills( "" ) << SomethingElse

I don't think so because a "normal" declaration of the operator<< would be
something like:
std::ostream& operator<<( std::ostream& os, CCharacter& CChar)

and so I would have to make it something like:
std::ostream& operator<<( std::ostream& os, const CSkill& Skill, const
std::string& Prefix )

But the compiler complains (no suprise):
error C2804: binary 'operator <<' has too many parameters

Any suggestions or do I just have to stick with the way I'm doing it?