No, most bed frames don’t provide the support a mattress needs to perform properly. The average bed frame only features one or two support beams or slats designed to support an additional mattress base, not a mattress itself. Placing your mattress on support beams will cause your mattress to sag and wear down rapidly, along with voiding its warranty.
The only bed frames you can put a mattress directly onto are platform beds with numerous slats placed close together. The slats on platform beds are similar to standard bed slats and are capable of supporting a mattress without an extra bed base.