Consider a function f and a list p={a,b,c}. I want to get a list of

f[a,a]
f[a,b]
f[a,c]
f[b,a]
f[b,b]
f[b,c]
f[c,a]
f[c,b]
f[c,c]

In real use, p can have higher dimensions, and f may take more arguments, e.g. I may need to generate a list of f[a,a,a,a] through f[z,z,z,z]. Is there a cleaner way to do this other than making lists that approximately repeat the elements of p then use MapThread? e.g.

p={a,b}
p1 = {a, a, b, b}
(* or: p1 = {p, p}\[Transpose] // Flatten *)
p2 = {a, b, a, b}
(* or: p2 = {p,p}//Flatten *)
MapThread[f, {p1, p2}]

Consider a function f and a list p={a,b,c}. I want to get a list of

f[a,a]
f[a,b]
f[a,c]
f[b,a]
f[b,b]
f[b,c]
f[c,a]
f[c,b]
f[c,c]

In real use, p can have higher dimensions, and f may take more arguments, e.g. I may need to generate a list of f[a,a,a,a] through f[z,z,z,z]. Is there a cleaner way to do this other than making lists that approximately repeat the elements of p then use MapThread? e.g.

p={a,b}
p1 = {a, a, b, b}
p2 = {a, b, a, b}
MapThread[f, {p1, p2}]

Here, a, b can be matrices