Yes, there absolutely was a definition - in fact, Marxists (and communists more specifically) were quite distinctive in actually producing contemporary theories of what fascism was and how to define it during the 1920s and 1930s. The issue was that this definition emphasised the capitalistic roots of the ideology, viewing fascism as the inevitable final stage of capitalistic development where the system relied on brute force to prolong itself in the face of crisis. There's a famous summary given by Georgi Dimitrov (and given at a world congress of the Third International, so very much reflected the 'official' Soviet line):

Fascism is neither the government beyond classes nor the government of the petty bourgeois or the lumpen-proletariat over the financial capital. Fascism is the government of the financial capital itself. It is an organized massacre of the working class and the revolutionary slice of peasantry and intelligentsia. Fascism in its foreign policy is the most brutal kind of chauvinism, which cultivates zoological hatred against other peoples.

The issue with this definition is that if fascism was the inevitable consequence of capitalism, anyone seeking to support a capitalistic system was supporting fascism, and were therefore fascist in effect, if not in name. As such, groups like social democrats, who helped prop up the capitalist system in the face of communist revolutionary efforts, became 'social fascists' in communist parlance. The idea was to force the working class to choose between revolution or fascist dictatorship, rather than seek to prolong the failing capitalist system.

There's a lot more detail on Marxist analyses of Nazism in this older thread from u/commiespaceinvader, and some discussions of alternative Marxist perspectives in this rather chaotic thread.