Drums work like speakers, only it's a kinetic impact rather than a magnetic impact that moves the material that moves the air.

All you need imagine is a drumstick striking the left side of the image rather than the magnetic coil shown in the image. That's how drums work.
Several things contribute to this.
The diameter of the drum. The bigger it is, the easier it is to produce low-frequency sound. Conversely, the smaller it is, the easier it is to produce high-frequency sound.
The capacity of the drum head to vibrate. If you think about it, the human ear isn't very good at picking up just one cycle of any sound. The fewer the number of cycles available to hear, the larger the amplitude must be to hear it.
The healthy, young auditory system can detect tones in quiet with frequencies ranging from approximately 20 to 20000 Hz. Figure 2-2 displays the standardized average thresholds for detecting tonal sounds of different frequencies when the sounds are approximately 500 milliseconds (ms) in duration. (Source)

This is why very large drums must be hit very hard to be easily heard.
- Third is the ability of the drum head to relax or return to its original position. The faster this can happen, the higher the frequency can be reproduced. If the drum head requires 0.1 ms to return to its original position, the drum itself cannot produce a frequency higher than 10 KHz. This is easily seen from the perspective of the speaker animation above. If the speaker cone can't return to its first position before the magnet sends the next pulse, the cone simply stays at its extended position and stops making sound. Likewise, the drum would stop making sound (all you'd hear is the sound of the drum stick whacking the drum head - the drum itself wouldn't make sound).
It's worth noting that the vibration and relaxation of the drum head are not instantaneous. They slow both in frequency and amplitude. The sound a person hears is not a single frequency, but a series of frequencies in a decaying pattern. The initial strike and the size of the drum create the initial, predominant frequency. It gets quieter and lower from there. This could influence the construction of an ultrasonic drum as the decaying frequency could enter the range of human hearing, although the amplitude may not be high enough for detection. (Useful Reference, see the part about "shimmer")
What is a drum?
Finally, it's worth noting that a drum from the perspective of a musical instrument is something that could be defined as "a musician hits a flexible drum head and the result is a sound not attributable directly to the object used to hit the drum."
A block of wood isn't a drum because what you're hearing is the sound of the drumstick hitting the block of wood, not the sound created by a flexible surface concussing against air. (Musicians don't refer to wood blocks as drums.)
Likewise, concrete isn't a drum because 99% of what you're hearing is the sound of the drumstick itself.
An upturned can is a drum because the metal, though much more ridged than a traditional drum head, is nevertheless concussing the air beneath it.
And steel drums are the best example of what might create an ultrasonic sound. But could one be created that dogs, but not humans, could hear?
An Ultrasonic Caribbean Steel Drum
From this, which has gotta be the end-all analysis of steel drums, we learn:
Another implication of using metal alloys for pan production is the effect of the material properties on the acoustics of the note. For instance, if a note were to be modeled as a circular clamped plate, its fundamental frequency can be expressed as [112]:
$$f_{0,0} = \frac{0.467t}{r^2}\sqrt{\frac{E}{\rho(1-v^2)}}$$
where t is the thickness of the note metal, r is the radius of the note up to the clamping boundary, $\rho$ is the density of the note metal, E is its Young’s modulus and v is Poisson’s ratio.
From page 29 of that document we get the variables filled and, using a steel thickness of 0.55 mm:
$$ f_{0,0} = \frac{1.2696}{r^2}$$
Which, if I've done the math right, gives us a drum head radius of 8mm to achieve a 20 Khz fundamental note.
Conclusion
The answer is no, you can't make a drum that emits a frequency beyond human hearing. The drum head's manufacture (a material with a quick enough recovery capability) and size (8mm and smaller using stainless steel) are physically too small for a creature of the size you're discussing to predictably hit with a drum stick (which must, itself, be smaller than the diameter of the drum head).