R, 84 81 bytes

Or R>=4.1, 74 bytes by replacing the word function with a \.

function(n){m=t(n^2)
while(T<n){T=nrow(m)
m=rbind(n^2-T:1-F,t(m[T:1,]))
F=F+T}
m}

Try it online!

Uses the common "rotate and add a row on top" approach.

R, 84 81 67 bytes

Or R>=4.1, 60 bytes by replacing the word function with a \.

Edit: -14 bytes thanks to @Giuseppe inspired by @Bubbler's Jelly answer.

function(n){m=t(1)
while(T<n)m=rbind(1:n,(T=nrow(m))+t(m[T:1,]))
m}

Try it online!

Uses the common "rotate and add a row on top" approach.

See also @Giuseppe's direct construction of the matrix.

R, 84 bytes

Or R>=4.1, 77 bytes by replacing the word function with a \.

function(n){m=t(s<-n^2)
while(F<s-1){a=nrow(m)
m=rbind(s-a:1-F,t(m[a:1,]))
F=F+a}
m}

Try it online!

Uses the common "rotate and add a row on top" approach.

Probably missed some trivial golf, but I'm looking at this for too long already...

R, 84 81 bytes

Or R>=4.1, 74 bytes by replacing the word function with a \.

function(n){m=t(n^2)
while(T<n){T=nrow(m)
m=rbind(n^2-T:1-F,t(m[T:1,]))
F=F+T}
m}

Try it online!

Uses the common "rotate and add a row on top" approach.