A right pyramid has a square base with area 288 square cm. Its peak is 15 cm from each of the other vertices. What is the volume of the pyramid, in cubic centimeters?
[asy]
import three;
triple A = (0,0,0);
triple B = (1,0,0);
triple C = (1,1,0);
triple D = (0,1,0);
triple P = (0.5,0.5,1);
draw(B--C--D--P--B);
draw(P--C);
draw(B--A--D,dashed);
draw(P--A,dashed);
label("$A$",A,NW);
label("$B$",B,W);
label("$C$",C,S);
label("$D$",D,E);
label("$P$",P,N);
triple F= (0.5,0.5,0);
label("$F$",F,S);
triple M=(B+C)/2;
draw(P--F--B,dashed);
[/asy]

Let $F$ be the center of the square base.  Since the pyramid is a right pyramid, triangle $PFB$ is a right triangle.  The area of the base of the pyramid is 288 square cm, so the length of each side of the base is $12\sqrt{2}$ cm.  Since $F$ is the center of the base, $FB$ is half the diagonal of the base, or $(12\sqrt{2}\cdot\sqrt{2})/2 = 12$ cm.  Applying the Pythagorean Theorem to triangle $PFB$ gives  \[PF = \sqrt{PB^2 - FB^2} = \sqrt{225 - 144} = \sqrt{81} = 9,\]so the volume of the pyramid is $[ABCD](PF)/3 = (288)(9)/3 = 288\cdot 3 = \boxed{864}$ cubic centimeters.