A solid electrolytic capacitor that contains a capacitor element including an anode body, a dielectric layer, and solid electrolyte is provided. The capacitor element also includes an anode lead (e.g., wire, tape, etc.) that is electrically connected to the anode body. A first portion of the anode lead is embedded within the anode body, while a second portion of the anode lead extends from the anode body in a longitudinal direction. Contrary to conventional capacitors in which the exposed portion of the anode lead is supported by a complex and bulky lead frame assembly, there is no lead frame present in the capacitor of the present disclosure. Thus, the volumetric efficiency of the finished capacitor can be increased. An assembly containing a matrix of multiple solid electrolytic capacitor elements is also provided, as is a method for forming a matrix that comprises multiple solid electrolytic capacitor elements.