Desmos 3D, 19 (+ \$\log_{256}(7\times2)\approx 0.475919\$) bytes

n=0
a=hsv(n,x,z
y=0

Input into \$n\$. Requires you to set the colour of the equation \$y=0\$ to the defined custom colour, as well as tick the Extend to 3D option, for an extra \$\log_{256}(7\times2)\approx 0.475919\$ bytes. The square is defined by \$0\leq x\leq 1,0\leq z\leq 1\$.

Try it here!

Scratch, 258 247 bytes

Highly suboptimal...

Try it here

define x(i
go to x:(-50)y:(-90
set pen(color v)to((i)/(3.6
set pen(brightness v)to(
repeat(256
set pen(saturation v)to(
repeat(256
pen down
pen up
[saturation
move(1)steps
end
set x to(-50
change y by(1
[brightness
define(c
change pen(c)by(.390625

For a bit of interactivity I customized it so that moving the cursor will redraw a new square, taking the normalized x-coordinate of the cursor as the input. Here is an example output, \$i=360°\$:

R, 98 bytes

\(n)image(matrix(seq(m<-outer(k<-0:255/255,k,Vectorize(\(a,b)hsv(n/360,a,b)))),256),c=m,ax=F,as=1)

Draw it on rdrr.io!
check byte count

JavaScript (ES6), 193 bytes

Generates a <canvas> of 256×256 pixels and fills it by converting from HSV to HSL. Most probably not the shortest approach.

h=>{document.write`<canvas id=c height=256>`;a=c.getContext`2d`;for(y=256;y--;)for(x=256;x--;a.fillRect(x,y,1,1))a.fillStyle=`hsl(${h} ${v=1-y/255,v-=l=v-v*x/510,v/(l<.5?l:1-l)*100} ${l*100})`}

(Don't) Try it online!

Graphical output

f=
h=>{document.write`<canvas id=c height=256>`;a=c.getContext`2d`;for(y=256;y--;)for(x=256;x--;a.fillRect(x,y,1,1))a.fillStyle=`hsl(${h} ${v=1-y/255,v-=l=v-v*x/510,v/(l<.5?l:1-l)*100} ${l*100})`}

document.write = _ => _; f(123)

<div style="float:left"><input type="range" min="0" max="360" value="123" style="writing-mode:vertical-lr;direction:rtl" oninput="f(this.value)" /></div><canvas id=c height=256>

R, 74 bytes

\(h)image(matrix(1:1e6,1e3),c=outer(j<-0:999/999,j,hsv,h=h/360),ax=F,as=1)

Try it at rdrr.io

Wolfram Language (Mathematica), 48 bytes

Image@Table[Hue[2#/a,j/a,1-i/a],{i,a=6!},{j,a}]&

Creates a 720x720 image using the Hue[h,s,b] builtin that expects its arguments (hue, saturation, and brightness in that order) normalized from 0 to 1. Some good fortune that 6!=720 is both shorter to code than 256 and also twice the 360 needed to normalize the hue argument. (Although I guess replacing 6! by 360 removes the need for the 2 and so yields the same code length.)

CSS, 102 bytes

div{--h:123;width:256px;height:256px;
background:linear-gradient(0,#000,transparent),hwb(var(--h)0 0)linear-gradient(90deg,#fff,transparent)
}

<div id=o></div><input type=number value=123 min=0 max=359 oninput=o.style.setProperty("--h",this.value);

The work is done by the second line of CSS in the snippet; the rest of the CSS makes the image visible and gives the input a default value for convenience; the HTML allows you to change the input's value. (Since CSS can't read input in any other way, I assume using a variable is legal.)